Structural properties of bipartite subgraphs

This paper establishes sufficient conditions that force a graph to contain a bipartite subgraph with a given structural property. In particular, let $\beta$ be any of the following graph parameters: Hadwiger number, Haj'{o}s number, treewidth, pathwidth, and treedepth. In each case, we show that there exists a function $f$ such that every graph $G$ with $\beta (G)\ge f(k)$ contains a bipartite subgraph $\widehat{G}\subseteq G$ with $\beta (\widehat{G})\ge k$.