Edge-colouring and orientations: applications to degree- and $\chi$-boundedness

TL;DR

This paper generalizes Ramsey's theorem for large minimum degree graphs, showing they contain large monochromatic or structured subgraphs, and explores how certain graph classes preserve degree-boundedness under extensions.

Contribution

It introduces new tools for analyzing degree-boundedness in graph classes and characterizes oriented graphs that preserve degree-boundedness when excluded.

Findings

Large minimum degree graphs contain large monochromatic induced subgraphs.

Certain graph classes like odd-signable and Burling graphs are degree-bounded.

Characterization of oriented graphs that preserve degree-boundedness when excluded.

Abstract

We prove a new generalisation of Ramsey's theorem by showing that every $2$-edge-coloured graph with sufficiently large minimum degree contains a monochromatic induced subgraph whose minimum degree remains large. From this, we also derive that every orientation of a graph with large minimum degree contains either a large transitive tournament or an induced antidirected digraph whose minimum degree is still large. As a consequence, we obtain two general tools showing that certain extensions of degree-bounded graph classes preserve degree-boundedness. A hereditary class $\mathcal{G}$ is {\it degree-bounded} if, for every integer $s$, there exists $d=d(s)$ such that every graph $G\in \mathcal{G}$ either contains $K_{s,s}$ or has minimum degree at most $d$. With these tools, we obtain for instance that odd-signable graphs and Burling graphs are degree-bounded. We also characterise exactly the oriented graphs $F$ such that the graphs admitting an orientation without any induced copy of $F$ are degree-bounded.