Structural properties of bipartite subgraphs

TL;DR

This paper identifies conditions under which graphs must contain bipartite subgraphs with specific structural properties, based on parameters like Hadwiger number and treewidth, providing bounds for their existence.

Contribution

It establishes sufficient conditions linking graph parameters to the guaranteed presence of bipartite subgraphs with large structural parameters.

Findings

Existence of functions f for each parameter ensuring bipartite subgraphs with large parameters

Conditions for graphs to contain bipartite subgraphs with specified structural properties

Generalization across multiple graph parameters such as Hadwiger number and treewidth

Abstract

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 $\beta(G)\geq f(k)$ contains a bipartite subgraph $\hat{G}\subseteq G$ with $\beta(\hat{G})\geq k$.