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

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