Highly Connected Graphs Have Highly Connected Spanning Bipartite Subgraphs

Abstract

For integers $k,n$ with $1 \le k \le n/2$, let $f(k,n)$ be the smallest integer $t$ such that every $t$-connected $n$-vertex graph has a spanning bipartite $k$-connected subgraph. A conjecture of Thomassen asserts that $f(k,n)$ is upper bounded by some function of $k$. The best upper bound for $f(k,n)$ is by Delcourt and Ferber who proved that $f(k,n) \le 10^{10}k^3 \log n$. Here it is proved that $f(k,n) \le 22k^2 \log n$. For larger $k$, stronger bounds hold. In the linear regime, it is proved that for any $0 < c < \frac{1}{2}$ and all sufficiently large $n$, $f(\lfloor cn \rfloor, n) \le 30\sqrt{c}n$. In the polynomial regime, it is proved that for any $\frac{1}{3} \le \alpha < 1$ and all sufficiently large $n$, $f(\lfloor n^\alpha \rfloor ,n) \le 9n^{(1+\alpha)/2}$.