Spanning bipartite subgraphs of highly connected graphs

Determine whether there exists a function f such that, for every positive integer k, every f(k)-connected graph contains a spanning bipartite subgraph that is k-connected.

Background

The paper discusses an old conjecture attributed to Thomassen concerning the preservation of vertex-connectivity when a graph is restricted to a spanning bipartite subgraph. The authors prove an asymptotic result: every k-connected graph with k growing faster than log n contains a spanning bipartite subgraph with connectivity of order k. This establishes a linear bound in the relevant asymptotic regime but does not settle the conjecture for every k independently of the graph order.

References

For the final application of the thinning--sprinkling technique, we return to the connectivity of graphs. An old conjecture of Thomassen from 1989 asks whether there exists a function $f$ such that, for every $k$, every $f(k)$-connected graph has a spanning bipartite subgraph which is $k$-connected.

— Thinning and sprinkling: from robust sampling to almost Hamiltonicity  (2609.30165 - Christoph et al., 24 Sep 2026) in Section 1, subsection “The thinning-sprinkling technique”