Exact cycle and wheel graph connection numbers determined with new proofs
Hamilton-connected cores and five cycle--wheel Ramsey numbers
Discrete Mathematics
Summary
This paper solves a puzzle about how many dots and lines you need to guarantee certain patterns appear in colored networks, focusing on cycles and wheel shapes. The authors prove exact answers for some specific sizes that were previously unknown using clever structural methods instead of checking all possibilities. They also classify some special colorings related to these patterns, confirming previous counts without exhaustive search. Their approach uses properties of connected paths and vertex cuts in the graphs.
What this means in practice
- •For network reliability engineers: Use precise structural bounds on cycle and wheel subgraphs to design communication networks resilient against node failures.
- •For graph algorithm developers: Guide the design of efficient algorithms for detecting critical substructures in large networks based on these structural classifications.
A theory result. No direct application yet.
Authors
Zehui Shao, Hanxin Jiang
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.