graph algorithm developers

Abstract: The famous Four Color Theorem (4CT) states that any planar graph can be colored with four colors, but how do we find a 4-coloring? In our previous paper [FOCS'26], we presented a simple combinatorial algorithm to 4-color planar graphs in $O(n\log n)$ time. We improve it here to $O(n)$ time, which is optimal. The algorithm from our previous paper is based on proving that every planar triangulation has a linear number of reducible configurations or obstructing cycles. The bottleneck for the algorithm is that we need $O(\log n)$ Kempe changes to extend a recursive 4-coloring to the reducible configurations. Each Kempe change can involve a constant fraction of the graph and takes linear time. Here we show that all $O(\log n)$ Kempe changes can be identified and implemented in $O(n)$ time. The idea is to spend $O(n)$ time to preprocess the graph so that we can identify and simulate each good Kempe change in $O(n/\log n)$ amortized time. Thus, we implement all $O(\log n)$ Kempe changes in $O(n)$ total time. Based on this, we implement the whole 4-coloring in linear time. We note that creating such a representation for fast identification and simulation of the right Kempe changes is a challenge of data structures, unlike the previous paper that builds on a new graph theoretic understanding of reducible configurations.

Abstract: It was recently shown (arXiv:2610.09618) that the infimum of the approximation ratios of deterministic constant-round LOCAL algorithms for minimum dominating set on planar graphs is 7. The $(7+\varepsilon)$-approximation given there runs in $T_0+O((c_0/\varepsilon)^κ)$ rounds with $κ=\ln 3/\ln(24/23)\approx 25.81$, and it is asked there whether this exponent can be substantially reduced. We replace the heaviest-edge star selection in its clustering step by a random-centre rule: every part becomes a centre independently with probability $p$, and every non-centre joins its heaviest centre neighbour. For planar graphs this gives $κ<5.89$ with $p=3/8$ (three fair coins per part and level) and $κ<6.7$ with $p=1/2$. An analysis that counts only the join edges cannot go below $5.74$. For every proper minor-closed class of density $a$, the new exponent is never worse, and with about $\frac12\log_2 a$ fair coins per part and level it is smaller by a factor that tends to $8$ as the density grows. This is optimal for the rule when $a$ is an integer and the class contains all $K_{a,n}$. The planar bounds hold verbatim on the torus, the projective plane and the Klein bottle. We then turn to the dependence on $\varepsilon$. On paths, every deterministic $r$-round algorithm for fractional domination, and every randomized one that always outputs a dominating set, has (expected) ratio at least $1+1/(r+4)$. So these approximations need $Ω(1/\varepsilon)$ rounds on every minor-closed class that contains all paths, and on paths this is tight up to an additive constant. On forests and on outerplanar graphs, with no bound on the degree, $O(1/\varepsilon)$ rounds suffice, so there the optimal number of rounds is $Θ(1/\varepsilon)$. The planar level bounds, the bounds on paths and the deterministic bound on forests are machine-checked in Lean 4 with Mathlib.

Abstract: The greedy algorithm for minimum 3-cut splits a global minimum-weight cut and then the cheaper of its two sides; it is a tight $4/3$-approximation. We show that it is exact, from every global minimum-weight cut, whenever some optimal 3-partition has two parts joined by total weight at least the weighted edge connectivity $λ$. This extends both theorems of Hochbaum and Shmoys from unweighted planar graphs to all weighted graphs, recovers the $4/3$ bound, and generalizes to minimum $k$-cut with a one-paragraph proof. The remaining hard instances have all pair weights below $λ$; for them we prove a merge lemma for Gomory-Hu sides. Finally, we give a 7-vertex graph with a unique Gomory-Hu tree whose unique optimal 3-partition contains no side of that tree, so the first stage of Kapoor's algorithm does not suffice on its own. The main theorem, the 4/3 bound, the merge lemmas and the counterexample are verified in the Lean 4 proof assistant.

Abstract: We study the sample complexity of testing $k$-colorability in the dense graph model. A graph is $\varepsilon$-far from being $k$-colorable if at least $\varepsilon n^2$ edge deletions are required to make it $k$-colorable. We prove that, for every integer $k\ge 2$, a uniformly random induced subgraph on \[ O\!\left(\frac{k\log k}{\varepsilon}\right) \] vertices is non-$k$-colorable with probability at least $2/3$ whenever the original graph is $\varepsilon$-far from $k$-colorable. The proof introduces a trial coloring process based on nested palette systems and iteratively selected pivots. This yields an upper bound of $O(k\log k/\varepsilon)$, which is no worse than the previous state of the art $O(k\log(1/\varepsilon)/\varepsilon)$ throughout the nonvacuous parameter range, and is an improvement whenever $1/\varepsilon$ is superpolynomial in $k$. Moreover, for every fixed $k$, our bound matches the known lower bound, thereby establishing the optimal $Θ(1/\varepsilon)$ dependence on $\varepsilon$ and, to the best of our knowledge, achieving this optimal dependence for the first time.

Abstract: We study distributed approximation of Minimum Dominating Set in a very general class of sparse graphs, extending near-optimal approximation results from the minor-free setting to bounded expansion. In more detail, for fixed $d$ and $\varepsilon>0$, we give a new deterministic constant-round LOCAL algorithm for Minimum Dominating Set on graphs of maximum average degree at most $d$ and bounded depth-one shallow-minor density. Its approximation ratio is $d+1+\varepsilon$, which is optimal up to $\varepsilon$ for every even integer $d\geq 2$. This improves on the recent results of Amiri and Siebertz by extending beyond proper minor-closed classes. From this algorithm, we derive almost-tight constant-round approximations for a number of graph classes. We get $(7+\varepsilon)$-approximations for planar graphs and graphs of bounded Euler genus, and $(5+\varepsilon)$-approximations for their triangle-free subclasses. This answers the $7+\varepsilon$ question of Wawrzyniak positively, and matches the recent results of Amiri and Siebertz. We also apply our algorithm to classes of bounded queue-number or stack-number, obtaining bounds that are tight up to $\varepsilon$. Additionally, we answer questions of Amiri and Siebertz about improving the dependence of the round complexity on $1/\varepsilon$ and obtaining polynomial local computation for fixed parameters, when comparing the output to the usual domination number.

Abstract: A graph is called Pfaffian if it admits a Pfaffian orientation. The Pfaffian recognition problem asks whether a given graph is Pfaffian. In 1989, Vazirani and Yannakakis showed that this problem is polynomial time equivalent to testing whether a prescribed orientation is Pfaffian, and that the problem belongs to $\mathsf{coNP}$. The complexity of the problem for general graphs has remained open since then. For bipartite graphs, polynomial time recognition algorithms were obtained by Robertson, Seymour, and Thomas in 1999, and independently by McCuaig in 2004. We show that testing whether a given orientation is Pfaffian can be reduced to polynomial identity testing. This gives a randomized polynomial time algorithm for Pfaffian recognition on general graphs, placing the problem in $\mathsf{coRP}$.

Abstract: We study the bipanchromatic number $χ_B$, introduced by Lalou et al. (Discrete Appl. Math. 391 (2026), 511-522) as the maximum number of colors in a panchromatic hypergraph coloring where every color appears at least twice. They show that merging uniquely occurring colors in a maximum panchromatic coloring gives the lower bound $β=χ_P-\lceilα/2\rceil \leq χ_B$, where $χ_P$ is the panchromatic number and $α$ is the minimum number of such colors among maximum panchromatic colorings. They conjectured that $χ_B=β$ and subsequently disproved it (arXiv:2610.08042). We identify the smallest counterexample, with six vertices and three hyperedges, both minimum possible. We show that the ratio bound $χ_B/β<2$ is asymptotically sharp and prove minimality results for the order and hyperedge count. For $3\leα<χ_P$ we show that every integer from $β$ to $χ_P-1$ occurs as $χ_B$ while $χ_P$ and $α$ are held fixed. Furthermore, we discover that the values $|V|,χ_P,α$ directly decide whether the conjectured equality holds except when $α=4$, where a matching reduction resolves the ambiguity. We derive the optimal universal lower bound $L$ depending only on $|V|,χ_P,α$ in the nontrivial regime $0<2α<|V|$. Finally, we prove $χ_B/L\le5/4$ and $χ_B/L\le6/5+1/(5L)$, with $5/4$ sharp and $6/5$ asymptotically sharp as $L\to\infty$.

Abstract: In the \emph{connected multicut problem for weighted trees}, the instance is an edge-weighted tree $T = (V,E,ω)$, a set of terminals $Γ\subseteq V$, and a set of pairwise ``conflicts'' between these terminals represented by edges of a conflict graph $\mathcal{C} = (Γ,\mathcal{F})$. A \emph{connected multicut} for this instance is a partition $\mathcal{M}$ of the vertex set $V$ into subsets called the \emph{components of $\mathcal{M}$}, where each component induces a subtree of $T$, contains at least one terminal, but does not contain any conflicting terminal pair. Two cost functions for multicuts are considered: in the min-sum variant, we minimize the total weight of the edges between different components of $\mathcal{M}$, and in the min-max variant, we minimize the maximum, over all components, of the total weight of edges leaving the component. In the first part of the paper, we focus on algorithms for the min-max variant, for which we give a $2$-approximation algorithm for the unweighted case, a $(2+ε)$-approximation algorithm for the general case, and a mildly exponential exact algorithm for the unweighted case. In the second part of the paper, we study algorithms that take advantage of the structure of the conflict graph. We first give a polynomial-time exact algorithm for the min-max variant when $\mathcal{C}$ is a clique (the variant referred to in the literature as the \emph{multiway cut}). Next, we generalize this result by providing polynomial-time exact algorithms, for both variants, for instances whose conflict graph has constant \emph{split complexity} --- a new parameter that refines the concept of neighborhood diversity previously studied in the context of parameterized complexity. This unifies and generalizes previous results for the min-sum variant when conflict graphs are complete multipartite graphs or unions of cliques.

Abstract: A coloring of the edges of a complete graph with $r$ colors is balanced if the edges among any $r+1$ vertices use all $r$ colors. Erdős and Gyárfás (1999) conjectured that for all $r\ge3$ the complete graph $K_{r^2+1}$ has no balanced $r$-coloring. They proved the conjecture for $r=3$ and $r=4$. We prove the conjecture for $r=5$, the first new case settled in over two decades. There is no balanced $5$-coloring of $K_{26}$, and since $K_{25}$ has one, $K_{25}$ is the largest complete graph admitting a balanced $5$-coloring. We design a reduce--dispatch--enumerate workflow, where every machine step produces an artifact that can be independently checked. First, we use a machine-checked (Lean 4) proof to reduce the conjecture to a single extremal statement about the color class with the fewest edges: no graph on $26$ vertices has between $55$ and $65$ edges while every $6$ of its vertices span between $1$ and $11$ edges. We then search on the sparse class instead of its dense complement, which makes enumeration up to isomorphism feasible. With classical stability theorems for $K_6$-free graphs, we dispatch the five near-extremal edge counts analytically. For the exhaustive enumeration on the remaining six edge counts, we design efficient encodings with SAT modulo symmetries and parallel cube-and-conquer. We also certify all eleven edge counts: each of the $12{,}296$ subproblems carries a clausal refutation checked by the independent verifier drat-trim, and all $273{,}503{,}938$ symmetry-breaking clauses are also validated against the vertex permutations that justify them.

Abstract: We show that for every graph $G$ and every positive integer $k$, $G$ contains either $k$ vertex-disjoint even cycles with no edges between distinct cycles, or a set of $O(k^2\log k)$ vertices whose closed neighborhood meets every even cycle of $G$. Moreover, we give a polynomial-time algorithm that outputs one of the two outcomes. As an application, we show that every $K_{1,t}$-free graph containing no induced packing of $k$ even cycles has tree-independence number $O(tk^2\log k)$, with a corresponding decomposition computable in polynomial time.

Abstract: It has been conjectured that $χ_{\mathrm{la}}(T)\leq\ell(T)+2$ for every tree $T$ of order at least three, where $χ_{\mathrm{la}}(T)$ is the local antimagic chromatic number and $\ell(T)$ denotes the number of leaves. We disprove this bound and show that no absolute constant can replace $2$ in an inequality of this form. Our approach uses degree-based intervals containing all possible induced vertex weights under bijective edge labelings, together with a separation criterion that forces selected internal vertex weights to be distinct. We first construct caterpillars $T_3$ and $T_4$ satisfying $χ_{\mathrm{la}}(T_3)=\ell(T_3)+3$ and $χ_{\mathrm{la}}(T_4)=\ell(T_4)+4$, respectively. We then construct, for every positive integer $r$, infinitely many pairwise nonisomorphic caterpillars with exactly $r$ internal vertices for which every bijective edge labeling is antimagic. Each such caterpillar $T$ satisfies $χ_{\mathrm{la}}(T)=|V(T)|=\ell(T)+r$. Consequently, $χ_{\mathrm{la}}(T)-\ell(T)$ is unbounded even within the class of caterpillars, and no absolute constant $C$ can satisfy $χ_{\mathrm{la}}(T)\leq\ell(T)+C$ for every tree $T$.

Abstract: Asymptotic dimension is a large-scale invariant of metric spaces that was introduced by Gromov (1993). A major open problem in coarse graph theory asks whether, for every fixed graph $H$, the class of graphs excluding $H$ as a fat minor has bounded asymptotic dimension. A weaker conjecture asserts the same conclusion for graphs excluding $H$ as an induced minor. We substantially expand the range of graphs $H$ for which these conjectures are known to hold. Specifically, we prove bounded asymptotic dimension for the following classes of graphs: graphs excluding a fixed planar graph as a fat minor; and graphs excluding a fixed string graph as an induced minor. All of our results also hold for the stronger notion of Assouad--Nagata dimension, a refinement of asymptotic dimension. These results have a number of applications, two of which we mention here. First they imply that an induced minor-closed class is $χ$-bounded if and only if it does not contain some Burling graph. Second they imply that, in graph classes excluding a fixed planar graph as an induced minor, treewidth is polynomial in maximum degree.

Abstract: We present compact SAT encodings and certifying exact solvers for boxicity and cubicity. The boxicity of a graph is the minimum dimension in which it is the intersection graph of axis-aligned boxes; cubicity is defined analogously using unit cubes. Grid-based formulations face a geometric discretization bottleneck, growing rapidly with the target dimension. We avoid this bottleneck by encoding the total order of interval endpoints in each coordinate. For an $n$-vertex graph and target dimension $d$, our boxicity encoding uses $O(n^2d)$ variables and $O(n^3d)$ clauses, both linear in $d$. Requiring the left- and right-endpoint orders to agree yields a cubicity encoding of the same asymptotic size. Satisfying models are decoded into independently checked geometric representations, while unsatisfiable instances used as lower-bound evidence carry mechanically verified DRAT certificates. In a controlled evaluation with 186 trials per encoding on standard and structured graph families, the order encoding solves 96.8% of trials, compared with 71.0% for the known grid formulation, and reduces mean PAR-2 time from 178.21 to 21.70 seconds. The pipeline also certifies exact values for several named graphs. We turn one such value into a reusable obstruction by proving that every Kneser graph $K(n,k)$ with $k\ge3$ and $n\ge2k+1$ contains an induced Desargues graph. The resulting lower bound, together with an explicit three-dimensional representation of $K(7,3)$, establishes $\operatorname{boxicity}(K(7,3))=3$. This disproves the conjecture of Caoduro and Sebo that $\operatorname{boxicity}(K(n,k))=n-k$ for all $k\ge3$ and $n\ge2k+1$.

Abstract: In a seminal paper, Kratsch and Spinrad [SODA 2003 and SIAM Journal on Computing 2006] studied several important problems in structural graph theory that anticipated two trends in fine-grained complexity, both from the upper- and lower-bound side: * From the upper-bound side, for each of these problems the authors give an intermediate running time strictly between $n^ω$ and $n^3$, where $ω<2.371177$ is the matrix multiplication exponent. When optimized, this running time can be made $\tilde{\mathcal{O}}(n^{(3+ω)/2})$. By now, many more problems can be solved in this running time: directed unweighted or node-weighted APSP, a variety of matrix products such as max-min and dominance, RNA Folding and many more. For some of these, the community has also developed tools suggesting that this intermediate running time is necessary. * From the lower-bound side, the authors give lower bounds of $n^{ω-o(1)}$, conditioned on three natural hardness hypotheses on the complexity of quantified variants of triangle detection that account for different types of hardness of such problems. Despite this, it has remained open for over 20 years which of these two pieces of evidence reflects the truth for the problems that Kratsch and Spinrad study: whether the intermediate running time is indeed necessary and stronger lower bounds hold, or whether the hardness evidence of Kratsch and Spinrad is essentially tight. In this paper, we resolve this question and show that the latter is the case: all problems admit an $\tilde{\mathcal{O}}(n^ω)$ time algorithm. As an additional contribution, we make partial algorithmic progress on further problems in structural graph theory of intermediate complexity, improving over previous work [Chiu and Lu, STACS 2022 and Information and Computation 2024][Spinrad, Discrete Applied Math 2004].

Abstract: We prove that the independence sequence of every sufficiently large forest is strictly barrier log-concave: whenever $i_{k+1}>i_k$, one has $i_k^2>i_{k-1}i_{k+1}$. Consequently, every common downward coefficient tilt is unimodal, with one order threshold valid simultaneously for all $0<t\le1$. As an application, we prove that every sufficiently large well-covered tree has a strictly log-concave independence polynomial. More generally, the latter conclusion holds for well-covered forests with sufficiently large matching number. Every sufficiently large forest with a palindromic independence polynomial is also strictly log-concave, with its modes at the central ranks. The proof establishes local coefficient-curvature estimates uniformly over forests of unbounded maximum degree and over activities tending to zero. Combined with the fixed-density theorem of Fang, Lu, Nevo, Yao, and Zheng, these estimates complete the strict log-concavity interval down to rank one through $\lfloor17α/25\rfloor$; the terminal fraction is inherited. The classical decreasing tail then yields the barrier property. A cumulative-coefficient corona theorem transfers barrier log-concavity of the base forest to full strict log-concavity of its leaf corona.

Abstract: The metric dimension of a connected graph is the least number of vertices whose distance vectors distinguish all vertices. We show that deciding whether the metric dimension of a graph is at most a given integer is NP-complete on graphs of treewidth at most two. As the problem is polynomial on trees, the graphs of treewidth one, this settles the treewidth threshold of its hardness, which was known to lie between two and twenty-four. The reduction is from satisfiability of formulas in which every literal occurs at most twice. To such a formula with $n$ clauses and $r$ variables it associates a graph $G_\varphi$ of treewidth at most two with $O((n+r)^2)$ vertices, whose metric dimension is $2(n+r)+5$ if the formula is satisfiable and $2(n+r)+6$ otherwise.

Abstract: Edge deletion problems ask for the fewest edges whose deletion gives a graph with a specified property. We study two such problems on block graphs. First making the graph claw-free, and second making it a line graph. Since every line graph is claw-free, the second problem is at least as difficult as the first. We show that on block graphs the two optimum values are always equal, and both are attained by the line graph of a forest. The main step is a completion argument, in which starting from any claw-free solution, we put back edges to get some nice structure inside each block still maintaining the claw-free property. This reduces both edge-deletion problems to the same problem of selecting vertices within the blocks of the input graph. We solve it by dynamic programming in $O(n+m)$ time, which also produces an optimal deletion set and a forest whose line graph is what remains after the completion argument. We also extend the method to $K_{1,r+1}$-free edge deletion for every fixed $r$.

Abstract: First defined by Bouchitté and Todinca, Potential Maximal Cliques (PMC) are a well studied notion in graph theory due to its numerous applications. The PMCs of an undirected graph $G$ are defined as the maximal cliques of the minimal triangulations of $G$. Their number can be exponential in the size of the graph. PMCs were crucial in the design of a polynomial-time algorithm which solves Maximum Weight Independent Set on $P_6$-free graphs (Grzesik, Klimošová, Pilipczuk, Pilipczuk, 2020). They also play a central role in the proposal of a polynomial-time algorithm to solve Largest Induced Subgraph of treewidth $\le t$ (Fomin, Todinca, Villanger, 2013). Last but not least, certainly one of the main application of PMCs is the existence of an exact dynamic-programming algorithm to determine the treewidth of a graph (Fomin, Kratsch, Todinca, Villanger, 2008) which uses its list of PMCs as an input. In fact, the enumeration of PMCs stands as the bottleneck for the computation of treewidth with this method: the PMC-enumeration step takes output-quadratic time (Bouchitté, Todinca, 2002) while the dynamic-programming one takes a time linear in the number of PMCs. This two-step procedure was proven to be among the most performing approaches in practice to determine treewidth. Answering a longstanding open question, we propose an algorithm to enumerate PMCs in output-linear time, {\em i.e.} a running time linear in the number of PMCs. This offers nice perspectives not only from the theoretical viewpoint - better running time for the PMC-based methods determining treewidth, for Minimum Fill-In and Largest Induced Subgraph of treewidth $\le t$ - but also from the practical side, with the possibility to propose top-performance implementation.

Abstract: A graph is \emph{elementary} if its edges can be coloured with two colours so that the two edges of every induced path on three vertices receive different colours. Elementary graphs are one of the two basic classes in the Chvátal--Sbihi decomposition of claw-free perfect graphs; they include all line graphs of bipartite multigraphs and all complements of bipartite graphs. We prove that every elementary graph $G$ satisfies $\ch(G)=χ(G)=ω(G)$, confirming for this class the conjecture of Gravier and Maffray that claw-free graphs are chromatic-choosable. The proof combines the Maffray--Reed description of elementary graphs as augmentations of line graphs of bipartite multigraphs, Galvin's theorem, an extremal (edge-maximal) choice of counterexample, and a Hall-type lemma describing how list colourings of a complete graph minus a matching can fail.

Abstract: Under the Strong Exponential Time Hypothesis (SETH), for any constant $\varepsilon > 0$, the #SAT problem (counting the number of satisfying assignments of a CNF formula) cannot be solved in $\mathcal{O}^*((2-\varepsilon)^n)$ or $\mathcal{O}^*((2-\varepsilon)^m)$ time, where $n$ is the number of variables and $m$ is the number of clauses. On the other hand, simple exhaustive search and the standard inclusion-exclusion give $\mathcal{O}^*(2^n)$-time and $\mathcal{O}^*(2^m)$-time algorithms, respectively. A natural line of research is to beat these SETH-based barriers for restricted versions, such as #$k$-SAT (where each clause contains at most $k$ literals) or #$d$-occ-SAT (where each variable appears in at most $d$ clauses). It is known that the $2^n$-barrier can be broken when $k$ or $d$ is any constant; yet, whether the $2^m$-barrier can also be broken in these cases remains unknown. We present two simple algorithms that overcome the $2^m$-barrier. First, for #$k$-SAT, we give an $\mathcal{O}^*(2^{m(1-1/O(k))})$-time algorithm, thereby breaking the $2^m$-barrier for any constant $k$. Second, for #$2$-occ-SAT (which also includes the problem of counting the number of edge covers in a graph as a special case), we provide an algorithm with runtime $\mathcal{O}^*(2^{mω/3})$, where $ω< 2.3714$ is the matrix multiplication exponent. Furthermore, we demonstrate that both algorithms generalize to the weighted counting setting while preserving the same runtime bounds. On the hardness side, we present a reduction from #Max-$d$-SAT to #$d$-occ-SAT, showing that for any $d\ge 3$, #$d$-occ-SAT admits no $\mathcal{O}^*((2-\varepsilon)^m)$-time algorithm for any constant $\varepsilon>0$, even on positive formulas, under the hypothesis that #Max-3-SAT on $n$ variables admits no $\mathcal{O}^*((2-\varepsilon)^n)$-time algorithm for any such $\varepsilon$.

Abstract: PATTERN, a synthetic node-classification benchmark still used to rank graph networks, plants a random subgraph in a random host graph and asks which nodes belong to it. Its generator draws 100 patterns once and plants the same ones in every split, so each test graph carries a pattern that also appears in 100 training graphs. A matcher that reads these patterns off the training labels and looks them up in each test graph reaches 99.2 class-weighted accuracy with no trained parameters, against 89.8 for the best published model. Inference under the generator without the patterns reaches 86.8. Two published models, CKGConv and GRIT, score lower on patterns they have not seen: on new graphs with fresh patterns both fall below this pattern-blind predictor, and withholding patterns from CKGConv's training lowers its accuracy on the test graphs that carry them. Training CKGConv with a new pattern in every graph raises its accuracy on fresh patterns by about 2 points, to the level of the pattern-blind predictor, at some cost on the reused patterns. The same kind of inference under the generator matches the best published result on CLUSTER and, without any training, scores above every published result we found on Minesweeper's released board. We recommend that synthetic benchmarks report scores on fresh templates beside reused ones, publish baselines built from their generators, and state how their metric is weighted and pooled.

Abstract: Recurrent algorithmic reasoners are trained with per-step hints on small graphs and run for many more steps on large ones. On breadth-first search in SALSA-CLRS, we find that a reasoner can reproduce every training hint and still fail on deep graphs: after some exact steps, it declares nodes reached before the search gets there. We call this a premature flood and trace two routes to it. In our GRU processor with a degree input, the flood is carried in the hidden state and set off when that input is in the tail of the training distribution. Setting the input higher at inference brings the flood sooner, and none of the six flooding models counts steps. Without the input, summing the neighbors' states instead of averaging them also breaks the processor. In the benchmark's own GIN and PGN, trained with its code, some seeds instead flood from a fixed step, on sparse and dense graphs alike. Clamping the degree input or adding 200 dense training graphs removes every failure of the GRU processor. Detaching the state between steps avoids the flood in that processor (0 of 23 seeds, against 6 under full backpropagation) but not with sums or a residual update, and two published remedies leave floods. None of the benchmark's test sets is both deep and dense, but its deepest one shows the second route: on its Delaunay graphs at 1,600 nodes, one GIN seed floods from step 18. We therefore recommend auditing the range of every input and aggregate, which catches the first route, and testing on deep graphs, which catches both.

Abstract: Norin and Steiner recently proved the Linear Hadwiger Conjecture. That is, there is an absolute constant $C$ such that every graph $G$ satisfies $χ(G)\leqslant C\,\text{had}(G)$, where $χ(G)$ is the chromatic number and $\text{had}(G)$ is the Hadwiger number of $G$. Their proof gives a non-optimised constant $C$ of order $10^{100}$. This paper uses extensive AI-based optimisation to show that every graph $G$ satisfies $χ(G) \leqslant 19{,}885{,}160\,\text{had}(G)$.

Abstract: A perfect Italian dominating function assigns a label from $\{0,1,2\}$ to each vertex so that the labels in the neighborhood of every zero-labeled vertex sum to exactly two. Although the associated decision problem (PID) is NP-complete on chordal graphs, the complexity status on split graphs remains open. We give an $O(n^4)$ time algorithm for split graphs on $n$ vertices. We note that this is in contrast to Roman, Italian, and perfect Roman domination, which are NP-complete on split graphs. We further show that this tractability boundary is tight: PID is NP-hard on graphs at deletion distance one from split graphs. We also provide dichotomy results for PID on restricted extensions of split graphs formed by attaching pendant stars, and establish inapproximability results for the same.

Abstract: We give the first polynomial improvements over the textbook algorithms for $3$SUM and All-Pairs Shortest Paths (APSP): we show how to deterministically solve $3$SUM on $n$ integers of polynomial size in $O(n^{1.9992})$ time and APSP on directed $n$-vertex graphs with polynomially bounded integer weights in $O(n^{2.9995})$ time. This refutes the $3$SUM and APSP hypotheses. Using known reductions, we also refute the real-valued versions of the $3$SUM and APSP hypotheses, the Exact Triangle hypothesis, the Zero-Weight $k$-Clique hypotheses, and the three rectangular hinted Online Matrix--Vector conjectures of van den Brand, Nanongkai, and Saranurak, and we give polynomial speedups for a variety of other problems. All of these results follow from a single new algorithm for thin matrix products. Let $X$ be an $N\times D$ integer matrix and $Y$ a $D\times N$ integer matrix with $D\le N^{1/18}$, and let $W$ be any set of at most $N^2/\sqrt D$ positions. We compute the entries $(XY)[I,J]$, $(I,J)\in W$, in $O(N^2/D^{0.063})$ operations, which is polynomially less than the time needed to write down $XY$ or to compute $N^2/\sqrt D$ inner products one by one. We design this algorithm by modifying a variant of Coppersmith's rectangular matrix multiplication algorithm, built from a ten-multiplication identity of Schönhage, to perform only the operations needed for the entries in $W$, and show that few operations are needed. Interpreted as a graph algorithm, this solves the All-Edges Sparse Triangle problem in truly subquadratic time on sparse lopsided tripartite graphs where two parts have $n$ vertices but one part has $n^{\varepsilon}$ vertices for $\varepsilon<0.12$. By known reductions, Exact Triangle, and hence $3$SUM and APSP, reduce to this problem. We also give a data structure version that answers queries for single entries of $XY$, not known in advance.

Abstract: Graph neural networks (GNNs) are the standard tool for node representation learning and are increasingly used in high-stakes settings. Their message-passing backbone, however, can amplify topological bias, raising fairness concerns. We study group fairness at the level of downstream predictions for node classification, link prediction, and node regression, and bound the demographic parity gap for an arbitrary number of sensitive groups. Our node classification bound is provably no looser than the closest prior result. For link prediction, ours is the first bound on the parity gap of the deployed sigmoid-activated prediction rather than a pre-activation proxy, and for node regression we provide the first such bound. Across all three tasks, the analysis identifies two distinct sources of bias: the separation between group means and the within-group covariance of the final representations. Building on this insight, we embed fairness into propagation itself by augmenting the convex smoothing problem underlying APPNP with a convex group-mean constraint and a within-group covariance regularizer. Unfolding projected gradient descent on this problem yields FairProp, whose layers pair a propagation step with a closed-form projection and which provably converges linearly to the unique fair optimum. Experiments on three tasks show that FairProp, even with exact group-mean equalization alone, provides a strong inductive bias that achieves excellent fairness-utility trade-offs against strong baselines.

Abstract: A Moore graph simultaneously asks for two things that normally pull in opposite directions: every vertex should have many neighbors, while every pair of vertices should remain close. For diameter two and degree $k$, the Moore bound is $k^2+1$, and equality forces a remarkably rigid graph: it is strongly regular with parameters $(k^2+1,k,0,1)$. Hoffman and Singleton showed that such a graph can exist only for $k=2,3,7$, and possibly 57. The degree-57 case on 3,250 vertices remains open. This paper is not an attempt to settle that open problem. Instead, we ask a more algorithmic question: How efficiently can the defining Moore structure itself be searched exactly? We begin from a human-developed exhaustive search based on the radius-two Moore tree. Leaves fall into $k$ groups of size $k-1$, and the edges between every pair of such groups form a perfect matching - a permutation. For efficiency, a partial construction is abandoned immediately when it creates a triangle or quadrangle. We then describe a sequence of exact algorithm-engineering improvements developed through a human-guided AI process and accepted only after mathematical checking and controlled regression tests. The main steps are reversible short-cycle forbid counters, bit-parallel domains, minimum-remaining-value branching, singleton propagation, matching-based AllDifferent filtering, balanced replayable parallel frontiers, residual-symmetry orbit pruning, and canonical deduplication of frontier states. On the principal $k=9$ benchmark, stronger symmetry cuts wall time from 47.6 to 17.9 seconds on a six-core laptop setup. Further parallelization speedup is reported separately; on a 16-core/32-thread machine, median wall time is reduced to 5.28 seconds. Degrees 10 and 57 are used as stress tests, and a five-minute degree-57 run uses only 161 MiB peak resident memory, showing that state size is modest.

Abstract: We refute the Strong Roberson Conjecture, which asserts that adding any graph outside a class closed under minors and disjoint unions strictly increases the distinguishing power of homomorphism counts from that class. More precisely, we construct connected graphs $H$ for which counts from graphs excluding $H$ as a minor determine the number of homomorphisms from $H$ to any target graph. We also refute the analogous conjecture with immersions in place of minors. We give explicit infinite families of excluded graphs, including cubic bipartite graphs that yield counterexamples for both relations. The proof introduces a method for deriving exact homomorphism count dependence from modular equivalences. We obtain these equivalences for infinitely many primes using prime-order automorphisms of graphs that exclude their orbit quotients as minors or immersions.

Abstract: We construct near-spanning balanced antidirected trees that fail to embed in highly symmetric tournaments despite the conjectured semidegree condition being satisfied. More precisely, for every sufficiently large odd integer $n$, we exhibit a regular vertex-transitive tournament on $n$ vertices and a balanced antidirected caterpillar on $n-1$ vertices with maximum degree at most $(1+o(1))n/\log_2 n$ that is not contained in the tournament. The host satisfies the strict $k/2$ threshold in both the semidegree and pseudo-semidegree settings, where $k$ is the number of arcs of the target tree. Consequently, this gives counterexamples to the uniform sublinear-degree interpretations of Conjectures~6.8 and~7.6 in Stein's survey (2024). The order $n-1$ of the target is best possible under the strict semidegree hypothesis. We also give a six-vertex balanced antidirected double-star missing from the seven-vertex Paley tournament, and show that seven is the smallest host order for a pseudo-semidegree counterexample with this fixed double-star.

Abstract: We give a randomized polynomial-time algorithm that colors any promised $3$-colorable graph on $n$ vertices with $\smash{O(n^{4/23}) = O(n^{0.17391\ldots})}$ colors, improving on the recent bounds of $O(n^{0.19539})$ by Bansal, Huang, and Lee and Narang and Tang who obtained $O(n^{(13-\sqrt{97})/18+ε})=O(n^{0.17506\dots + ε})$ colors for every fixed $\smash{ε>0}$. To prove our result, we start from a fixed-level semidefinite relaxation, where we use a finite-depth recursion on Gaussian covers. Fixing a root vertex, we group vertices by correlation with the root vector. Here, each step extends a cover of directions by one edge and transfers it to a successor group. Our key analytic ingredient is a variance bound for Gaussian maxima: for a maximum of $m\geq 2$ centered linear forms with coefficient norms at most $r$, mean $μ$, and variance $v$, we prove $v\leq r^2-μ^2/(2\log m)$ using Chen's Gaussian convexity theorem. Together with a variance-scale lower-tail estimate, this controls the threshold loss at each extension, which shows that root-conditioned vector colorings can either extract a large independent set from a group or bound its size, forcing a contradiction after constantly many steps. The resulting sparse-case guarantee combines with the dense progress bound of Kawarabayashi, Thorup, and Yoneda, and the recursion's numerical inequalities are verified via rational interval arithmetic.

Abstract: Let $K_t$ be obtained by attaching a pendant path of length $t$ to a fixed rooted graph $K$. We prove that $\mathrm{Mo}(L(K_t))-\mathrm{Mo}(K_t)$, the difference between its line-graph and original Mostar indices, is exactly affine on each parity class beyond a sharp uniform cutoff. The slope depends on distance-level neighbor counts and is positive for every non-bipartite core. Triangle chains give infinitely many graphs with $\mathrm{Mo}(L(G))>\mathrm{Mo}(G)$ at every positive cyclomatic number $c$, with maximum degree three, resolving Alex-Indulal Problem 3.3. At zero slope, a cactus branch-mass formula decides equality; identical vertex profiles need not give identical intercepts. Fixed cores giving equality for all sufficiently long attachments exist exactly when $c$ is odd. Maximum degree three suffices, and at every even $c$ a binary-tree construction gives equality on one eventual parity class.

Abstract: We study \textsc{Connected Dominating Set} on graphs whose closed-neighborhood set systems are $d$-semi-ladder-free. This structural condition strictly generalizes the biclique-free setting and provides a natural regime for connectivity-constrained domination. We obtain both a fixed-parameter algorithm and an approximate kernelization framework for the problem on this class. Our algorithmic result is based on a new compact representation theorem for inclusion-wise minimal set covers in $d$-semi-ladder-free set systems. Although the number of minimal set covers of size at most $k$ may be as large as $n^{Ω(k)}$, we show that all such set covers can nevertheless be encoded by a family of at most $k^{kd+1}$ tuples, and that this family can be enumerated in time $\Oh(k^{kd+2}\cdot nm)$. Combining this representation with a \textsc{Group Steiner Tree} subroutine, we obtain an algorithm for \textsc{Connected Set Cover}, which in turn yields an algorithm for \textsc{Connected Dominating Set} running in time $k^{kd+2}\cdot 2^k \cdot n^{\Oh(1)}$ and polynomial space. For the preprocessing result, we introduce grouped domination cores and dominator cores, and prove polynomial upper bounds on their sizes in $d$-semi-ladder-free graphs. Using these structures, we obtain, for every fixed $d$ and $\varepsilon>0$, a polynomial-time $(1+\varepsilon)$-lossy compression for \textsc{Connected Dominating Set} to an equivalent reduced instance of size $k^{\Oh(d^2/\varepsilon)}$. The reduced instance is a \textsc{Connected Dominating Set} instance on a $(d+2)$-semi-ladder-free graph.

Abstract: Let $W_s=K_1+C_{s-1}$ denote the wheel on $s$ vertices. We give structural proofs that $R(C_{14},W_{11})=27$ and $R(C_{15},W_{11})=29$. Together with the theorem of Chen et al. for $n\ge16$, these equalities give $R(C_n,W_{11})=2n-1$ for every $n\ge14$. The two boundary values were included in an earlier survey announcement. We also give structural proofs of $R(C_8,W_7)=15$, $R(C_9,W_7)=17$, and $R(C_8,W_9)=15$. The common starting point is a Hamilton-connected core lemma. For the eleven-vertex wheel, bounds on vertex connectivity and on the matching number of a bipartite graph associated with a local cycle yield a vertex cut of order nine. Paths with prescribed endpoints then rule out every possible pair of orders of the two remaining vertex sets. For the smaller wheels, we use the structure of critical cycle colorings and local cycle-shortening arguments. We also give complete structural classifications of the $(C_8,C_6)$- and $(C_9,C_6)$-critical colorings, recovering the previously reported counts 24 and 26. All proofs are combinatorial and use no exhaustive graph enumeration.

Abstract: A graph is $K_3$-irregular if its vertices belong to pairwise distinct numbers of triangles. We prove that no $8$-regular $K_3$-irregular graph exists, settling the last unresolved case. Following the initial discovery of such graphs for regularities $r \in \{10,11,12\}$ (Stevanovi'c et al., 2024), our previous work (Hak et al., 2025) showed that no such graphs exist for $r \le 7$, provided the first example for $r=9$, and proved that any $8$-regular candidate must have between $17$ and $22$ vertices. We exclude these possible orders for $r=8$ by combining careful analysis of triangle degrees with integer linear programming techniques. Meanwhile, a recent construction (Zhang, 2026) established that regular $K_3$-irregular graphs do exist for all $r \ge 9$. Together with our results, this establishes that an $r$-regular $K_3$-irregular graph exists if and only if $r\geq 9$.

Abstract: We explore the frontier beyond the recently discovered barrier represented by the \emph{quantum automorphism group} $qut(M)$ in the classification theory of planar graph homomorphisms $PlGH(M)$. We show that analyzing the spectral relations of $M$ can prove \#P-hardness when traditional vertex separation and domain-reduction methods with planar edge gadgets provably fail due to the $\qut(M)$ barrier. We prove two criteria of \#P-hardness for $PlGH(M)$: a spectral criterion and a determinant criterion. It is known that the core problem for the classification of $PlGH(M)$ for nonnegative matrices $M$ is for positive definite entry-wise positive matrices. We use the spectral criterion to show that $PlGH(M)$ is \#P-hard for all circulant matrices of prime order $q \ge 3$, while for $q=2$ it is precisely the matchgate case and is P-time computable by the FKT algorithm (for planar perfect matching). We also prove a complexity dichotomy for $\PlGH$ problems defined by tensor products of 2 by 2 matrices. This gives a complete complexity classification for this class of matrices, and the FKT algorithm together with a holographic transformation is \emph{universal}---every $PlGH(M)$ is either (1) P-time computable over all graphs, or (2) \#P-hard in general but P-time computable over planar graphs, or (3) \#P-hard over planar graphs; furthermore, $PlGH(M)$ in (2) consists of precisely those computable by FKT with a holographic transformation.

Abstract: A polychromatic $2$-coloring of a triangulation is a $2$-coloring of the vertices such that no face is monochromatic. The discrepancy of a coloring is the maximum difference between the sizes of the color classes. Asayama and Matsumoto (Graphs and Combinatorics, 2022) proved that every triangulation admits a polychromatic $2$-coloring with discrepancy at most $\tfrac{5n-16}{9}$, and that there exists a class of triangulations for which every polychromatic $2$-coloring has discrepancy at least $\tfrac{n}{3} - 2$, where $n$ is the number of vertices. We improve the upper bound, showing that every triangulation admits a polychromatic $2$-coloring with discrepancy at most $\tfrac{3n-16}{7}$ and such a $2$-coloring can be computed in quadratic time. We also show a discrepancy of at most $n-\tfrac{4M}{3}$ for triangulations with a matching of size $M$. This implies, for example, that Delaunay triangulations admit a discrepancy of at most $\tfrac{n}{3}$. We provide a linear-time algorithm to compute a $2$-coloring whose discrepancy is at most $\tfrac{5n-24}{7}$. One of our results shows that any proper four coloring with the largest color class of size $\frac{n}{2}$ would imply a $2$-coloring with discrepancy at most $\frac{n}{3}$. The existence of such a proper coloring has been recently confirmed by Kawarabayashi, Yoneda, and Yoneda (arXiv 2026). Therefore the two results together confirm the discrepancy of at most $\frac{n}{3}$ for triangulations.

Abstract: The study of preordered semirings is motivated by applications in computer science, graph theory, and information theory, and provides tools for understanding the asymptotic preorder, which compares large powers of a pair of elements. This paper studies sequences which behave approximately as sequences of powers, but are not necessarily equivalent to geometric sequences. Our main result is that preordered semirings admit completions where such sequences, that we call approximately geometric, become equivalent to geometric sequences, and that existing characterizations of the asymptotic preorder extend to the completion. We provide several classes of examples of approximately geometric sequences in the semiring of tensors, and in the semiring of graphs. As a concrete application, we determine the strong converse exponent for binary hypothesis testing with composite Markov hypotheses.

Abstract: In this paper, we consider the complexity of enumerating inclusion minimal defensive alliances. We present a polynomial-delay algorithm on graphs with maximum degree 5. We complement this result by proving that there is no output-polynomial algorithm on bipartite graphs with maximum degree 6 and degeneracy 2, unless P = NP. Furthermore, there is an FPT-delay algorithm when parameterized by neighborhood diversity. This result is the first exploit of a recently published enumeration algorithm for ILPs. By way of contrast, we prove that, for the parameter pathwidth, there is no FPT-delay algorithm for enumerating all inclusion minimal defensive alliances (unless FPT = W[1]). To the best of our knowledge, this is the first non-enumerability result using parameterized complexity for variations of Another/Next-problems.

Abstract: In the vertex cover interdiction problem, we are given an undirected graph $G=(V,E)$, two integers $t$ and $k$ and a vertex subset $B\subseteq V$, and we are asked to find a set $X \subseteq B$ with $|X|\leq t$ such that $X$ hits (i.e., intersects) all the vertex covers of $G$ of size at most $k$. Recently, Grüne and Wulf proved that the problem is $Σ_2^p$-complete. However, their reduction relied on the fact that the vertex cover problem is NP-complete. This, in turn, means that we do not know the complexity status of the vertex cover interdiction problem when the input graph is restricted to a bipartite graph since the vertex cover problem can be solved in polynomial time for bipartite graphs. One of our main results shows that the vertex cover interdiction problem is NP-complete for bipartite graphs. In contrast, when $k$ is restricted to the minimum vertex cover size, i.e., we are only required to hit all the minimum vertex covers, we show that the vertex cover interdiction problem can be solved in polynomial time for bipartite graphs. This motivates us to study the parameterized complexity of the vertex cover interdiction problem for bipartite graphs when the difference of $k$ and the minimum vertex cover size is taken as a parameter. With this parameter, we show that the problem is $\mathrm{W}[1]$-hard, but can be solved in polynomial time when the parameter is constant (i.e., in XP time). We also show that the problem is fixed-parameter tractable when parameterized by $k$.

Abstract: We prove two main algorithmic results in spectral discrepancy. First, we give a deterministic polynomial-time rounding theorem for rational positive semidefinite matrices of arbitrary rank. The algorithm starts from any rational fractional signing and assigns one sign per original matrix. Its discrepancy is less than $3.37\,\|\sum_i \mathrm{Tr}(A_i)A_i\|^{1/2}$. This yields Kadison--Singer half-partitions with error below $1.69\sqrt{\varepsilon}$, as well as deterministic graph signings that control signed adjacency and signed degrees simultaneously. The proof builds on the spectral-potential method of Ezeunala and Jiang (2026) and introduces a new way to choose rounding directions. We prove polynomial bit complexity for the rounding procedure. Second, we give a Las Vegas algorithm for the Bilu--Linial signing problem on an arbitrary prescribed graph. If $G$ has $n$ vertices and maximum degree $Δ\ge3$, the algorithm terminates almost surely. It uses fewer than $100n^3$ insertion attempts in expectation and returns a signing with $\|A_s\|<2\sqrt{2(Δ-1)}$. For bipartite graphs its one-sided form gives the sharp universal bound $\|A_s\|<2\sqrt{Δ-1}$. The algorithm builds the signing by inserting vertices and recursively deleting and restoring neighbors after rejected insertions. In the analysis, the $\sqrt2$ gap to the Bilu--Linial conjecture comes from a factor of two in the bound for vertex deletions in the two-sided case. On a $d$-regular bipartite Ramanujan base the same signing produces a Ramanujan $2$-lift of that prescribed base.

Abstract: Let $A_1,\ldots,A_N$ be positive semidefinite matrices of rank at most $r$, with $\sum_iA_i=I$ and $\|A_i\|\le\varepsilon$. We prove that the original matrices admit signs with discrepancy $O(\sqrt\varepsilon\log(2r))$, independently of their dimension and number which is a significantly stronger result than what was known existentially. We give a deterministic algorithm with polynomial real-arithmetic work, and a separate existence proof requiring no computational assumptions. This extends our companion paper on rank-one Kadison--Singer discrepancy. A concave matrix power interpolates between the trace source, which pays a factor $r$, and the sandwich source, whose density response is harder to control. We prove that source concavity controls this additional response in the same inverse-Sylvester metric as the optimized spectral potential. As an application, a single spanning tree can be chosen simultaneously $O(\varepsilon\log^2(2s))$-spectrally thin for $s$ positive edge weightings of a common graph, provided every edge has leverage at most $\varepsilon$ in every weighting. The reduction preserves one common selection decision per edge. For incidence matrices with at most $t$ ones in every row and column, the diagonal specialization gives a deterministic walk on fractional colorings with discrepancy $O(\sqrt t\log(2t))$. The local-walk mechanism gives both existence and an efficient construction without using the Lovász local lemma. A Lean formalization of our existence proof has been completed and will be released shortly.

Abstract: The classical Weisfeiler-Leman algorithm (also known as the $2$-dimensional Weisfeiler-Leman algorithm) is a simple combinatorial algorithm that was originally designed as a heuristic for the graph isomorphism problem. However, it has also numerous connections to other areas such as algebraic graph theory, logics, proof complexity, combinatorial optimization and machine learning. We prove that the classical Weisfeiler-Leman algorithm terminates after $5(n-1)$ iterations. This improves over the previous best upper bound of $O(n \log n)$ by Lichter, Ponomarenko and Schweitzer [LICS 2019], and asymptotically matches the known lower bound of $Ω(n)$ by Fürer [ICALP 2001]. Additionally, building on our results for the $2$-dimensional case, we obtain an improved upper bound of $O(n^{k-1}/(k-2)! + n^{k-2})$ on the number of iterations performed by the $k$-dimensional Weisfeiler-Leman algorithm, for every $k \geq 3$. Our arguments actually hold for a larger class of sequences of colorings of $k$-tuples; in this larger class our upper bounds are essentially tight for all $k \geq 3$.

Abstract: We show that every planar graph with no $t \times t$ grid minor has treewidth at most $4t +4$. This improves on the previously best known bound of $\frac{9}{2}t - \frac{11}{2}$, due to Gu and Tamaki (2012), and is within a factor $2$ of optimal. A key step in the proof is showing the following result, which might be of independent interest: Every $2$-connected plane graph $G$ with radius $d$ and faces of size at most $k$ has a tree-decomposition of width at most $\max\{3d+ k+5, 2d+2k+1\}$ such that the vertex set of every face of $G$ is contained in some bag.

Abstract: We study the minimum coefficient degree $N_{k,\F}(G)$ of a Nullstellensatz certificate for Bayer's $k$-coloring equations, where the characteristic of $\F$ does not divide $k$. If $J$ is a \HJ\ join of non-$k$-colorable graphs $G,H$ and $m=\max\{N_{k,\F}(G),N_{k,\F}(H)\}$, then $N_{k,\F}(J)\leq m+k$. When deletion of the selected edge makes each input $k$-colorable, we also have $N_{k,\F}(J)\geq m$; the degree congruence then gives $N_{k,\F}(J)\in\{m,m+k\}$. This partially answers a question of Li, Lowenstein, and Omar. For three-coloring over $\F_2$, we construct an infinite $4$-critical family of exact degree seven, attaining the bound at input degree four. In contrast, every graph constructed from $K_4$ solely by \HJ\ joins has degree $O(\log n)$ and a certificate with polynomially many terms: joins preserve treewidth at most three, and balanced separators yield low-degree certificates. Additional vertex identifications are excluded from this obstruction. We classify all single identifications of the $25$-vertex base graph; exactly $36$ preserve degree seven, producing $24$-vertex $4$-critical graphs of treewidth four. A compressed self-join at adjacent true twins prevents degree loss and gives a repeatable rule adding four vertices per round. The rule does not establish degree amplification or preservation of criticality. Exact witnesses and standalone verification programs accompany the finite results.

Abstract: We study a graph simplification problem introduced by Bernstein, Bodwin, and Wein [ITCS'24]. We start with a graph with arbitrarily large positive edge weights and the goal is to reweight the edges to small aspect ratio (ratio between largest and smallest weight) while preserving the shortest paths structure (the sequence of vertices and edges along shortest paths). They studied whether polynomial aspect ratio is always possible. They proved that for general graphs, both directed and undirected, it is not: there exist graphs for which any shortest-paths preserving reweighting requires exponential aspect ratio. In contrast, they showed that every DAG (directed acyclic graph) admits a reweighting with linear aspect ratio. However, the resulting edge weights are not integers. This motivated them to pose the open question of whether all DAGs admit a reweighting with polynomially-bounded integer edge weights. Our main result is to answer this question in the negative: we prove that there exist DAGs for which any shortest-paths preserving integer reweighting requires weights of size $2^{Ω(n)}$. In fact, this is even true when the DAG has very simple structure: 3 layers of vertices with only 3 vertices in the middle layer. In contrast, we show that if the number of vertices in the middle layer is decreased to 2, then a linear upper bound is possible. We extend our exponential lower bound to the approximate version of the problem where only a single $α$-approximate shortest path in the original graph must be preserved as an exact shortest path in the reweighted graph. Our exponential lower bound holds even for any finite approximation ratio $α>1$.

Abstract: Let $λ$ be the minimum cut value of an $n$-vertex undirected multigraph. For every fixed $α>1$, we prove that there are $O(n^{\lceil2α\rceil-1})$ cuts of size strictly below $αλ$. The exponent is sharp. The proof combines splitting off and sampling with a bound on the size of nested families of vertex sets.

Abstract: The independent set polytope, or stable set polytope, of a graph $G$ is the 0/1-polytope defined by the convex hull of the characteristic vectors of all independent sets of $G$. We present a simple algorithm for computing a Hamilton path on the independent set polytope of a given $n$-vertex graph $G$ with amortized delay $\mathcal{O}(n)$. The independent sets are listed such that two consecutive sets differ either in removing a vertex, or adding a vertex and removing its neighbors from the independent set, i.e., the symmetric difference between two consecutive independent sets induces a star in $G$. As applications of this result, we obtain an algorithm to compute a Hamilton path on the matching polytope of an $m$-edge graph $G$ with worst-case delay $\mathcal{O}(m)$, which lists all matchings of $G$ in such a way that the symmetric difference between two consecutive matchings is a path on at most three edges. Furthermore, we obtain an algorithm to compute a Hamilton path on the chain polytope and order polytope of an $n$-element poset $P$ with amortized delay $\mathcal{O}(n)$, which lists all antichains of $P$ or all ideals of $P$, respectively, by star exchanges. Our algorithms are derived from the generic framework proposed by Merino and Mütze (FOCS'23+SICOMP'24) for computing Hamilton paths on arbitrary 0/1-polytopes, which uses a linear optimization procedure as a black box. Our algorithms bypass solving the computationally intractable maximum weight independent set problem by a simple and purely combinatorial greedy rule.