Strengthening topological colorful results for graphs

TL;DR

This paper generalizes topological coloring theorems to identify large bipartite subgraphs in properly colored graphs, leading to new insights on chromatic and circular chromatic numbers and their relation to Hedetniemi's conjecture.

Contribution

It extends three existing theorems in topological graph coloring, providing broader conditions for the existence of large bipartite subgraphs and new classes of graphs satisfying key coloring conjectures.

Findings

Generalized three theorems on topological coloring

Identified new graph families with equal chromatic and circular chromatic numbers

Established cases where Hedetniemi's conjecture holds for circular chromatic number

Abstract

Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind, respectively due to Simonyi and Tardos (Combinatorica, 2006), Simonyi, Tardif, and Zsb\'an (The Electronic Journal of Combinatorics, 2013), and Chen (Journal of Combinatorial Theory, Series A, 2011). As a consequence of the generalization of Chen's theorem, we get new families of graphs whose chromatic number equals their circular chromatic number and that satisfy Hedetniemi's conjecture for the circular chromatic number.