Structural properties of bipartite subgraphs

Abstract

This paper establishes sufficient conditions that force a graph to contain a bipartite subgraph with a given structural property. In particular, let β be any of the following graph parameters: Hadwiger number, Hajós number, treewidth, pathwidth, and treedepth. In each case, we show that there exists a function f such that every graph G with β(G) f (k) contains a bipartite subgraph Ĝ ⊆ G with β( Ĝ) k.

Structural Properties of Graph Products

“…Together with Lemma 19, Corollary 31 demonstrates that tw(G) and tw(G × K 2 ) are tied. Similarly, for pathwidth, as explained in our companion paper [38], an analogous result to Theorem 30 is implied by the excluded forest minor theorem [3,56].…”

“…We make use of the following lemma from our companion paper [38] which generalises the result by Erdős [28] which states that every graph contains a bipartite subgraph with at least half of the edges. Lemma 37 ([38]).…”