Bipartite Graphs Are Not Well-Quasi-Ordered by Bipartite Minors

TL;DR

This paper demonstrates that the class of bipartite graphs is not well-quasi-ordered under the bipartite minor relation by constructing infinite incomparable sets and specific pairs of graphs with different minor relations.

Contribution

It provides the first counterexamples showing bipartite graphs are not well-quasi-ordered by bipartite minors, clarifying the structure of bipartite graph minors.

Findings

Infinite set of pairwise incomparable bipartite graphs.

Existence of pairs where one graph is a bipartite minor but not a minor.

Existence of pairs where one graph is a minor but not a bipartite minor.

Abstract

In "Bipartite minors" [Journal of Combinatorial Theory, Series B, 2016], Chudnovsky et al. introduced the bipartite minor relation, a quasi-order on the class of bipartite graphs somewhat analogous the minor relation on general graphs and asked whether it is a well-quasi-order. We answer this question negatively by giving an infinite set of 2-connected bipartite graphs that are pairwise incomparable with respect to the bipartite minor relation. We additionally give two sets of infinitely many pairs of bipartite graphs: one set of pairs G, H such that H is a bipartite minor, but not a minor, of G, and one set of pairs G, H such that H is a minor, but not a bipartite minor, of G.