On a Conjecture of Thomassen

TL;DR

This paper investigates Thomassen's conjecture on the existence of highly connected bipartite subgraphs in sufficiently connected graphs, proving it holds approximately within a logarithmic factor.

Contribution

The paper provides the first partial proof of Thomassen's conjecture, establishing the existence of such subgraphs up to a logarithmic factor in connectivity.

Findings

Thomassen's conjecture is true up to a log n factor.

A spanning bipartite k-connected subgraph exists in sufficiently connected graphs.

The result advances understanding of connectivity and bipartite subgraph structures.

Abstract

In 1989, Thomassen asked whether there is an integer-valued function f(k) such that every f(k)-connected graph admits a spanning, bipartite $k$-connected subgraph. In this paper we take a first, humble approach, showing the conjecture is true up to a log n factor.