An $O(k\log(n/k))$ Bound on Spanning Bipartite Connectivity

2026-09-19

AI Summary

本文解决了图论中关于存在跨越二部k-连通子图的最小s-连通图问题，通过数学方法证明了f(k,n)=O(k log(n/k))的新上界。

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, $f(k,n)=O(k\log(n/k))$.

Problem

• spanning bipartite subgraph
• connectivity
• graph theory

Innovation

• spanning bipartite k-connected subgraph
• improved upper bound
• O(k log(n/k))