Highly Connected Graphs Have Highly Connected Spanning Bipartite Subgraphs

For integers k, n with 1 k 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) 10^10k³ n. Here it is proved that f (k, n) 22k² n. For larger k, stronger bounds hold. In the linear regime, it is proved that for any 0 < c < 12 and all sufficiently large n, if k= cn, then f (k, n) 30c n 30n (k+1). In the polynomial regime, it is proved that for any 13 < 1 and all sufficiently large n, if k = n^, then f (k, n) 9n^ (1+) /2 9n (k+1).