Improved connectivity bounds for spanning bipartite subgraphs in networks
An $O(k\log(n/k))$ Bound on Spanning Bipartite Connectivity
Discrete Mathematics
Summary
This paper looks at how well connected a network must be to ensure it contains a special kind of sub-network called a spanning bipartite k-connected subgraph. The authors improve previous mathematical bounds on the minimum connectivity needed. Their result shows you can guarantee this kind of sub-network exists if the network’s connectivity grows roughly like k times the logarithm of the network size divided by k. This refines earlier results and gets closer to a long-standing conjecture.
What this means in practice
- •For network designers: Design networks with guaranteed robust bipartite structures by understanding minimum connectivity requirements for fault tolerance.
- •For distributed systems engineers: Ensure the existence of certain resilient communication subgraphs in distributed systems for improved reliability planning.
A theory result. No direct application yet.
Authors
G. Gutin, Y. Hao, Y. Zhou
Abstract
For integers $1\le k\le n/2$, let $f(k,n)$ be the least integer $s$ such that every $s$-connected graph on $n$ vertices contains a spanning bipartite $k$-connected subgraph. Thomassen conjectured that $f(k,n)$ is bounded by a function of $k$ alone. Delcourt and Ferber proved $f(k,n)=O(k^3\log n)$, and Yuster subsequently obtained $f(k,n)\le22k^2\log_2 n$. We prove that, for $2\le k\le n/2$, \[ f(k,n)\le\min\left\{n-1,\, \left\lfloor6(k-1)\log_2\frac{n}{k-1}\right\rfloor\right\}. \] In particular, $f(k,n)=O(k\log(n/k))$.