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

For integers $1\le k\le n\mathrm{/}2$, 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 $f(k,n)\le 22k^2{\log }_{2}n$. We prove that, for $2\le k\le n\mathrm{/}2$, $$f(k,n)\le \min \{n-1,\lfloor 6(k-1){\log }_2\frac{n}{k-1}\rfloor \}\mathrm{.}$$ In particular, $f(k,n)=O(k\log (n\mathrm{/}k))$.