Induced subgraphs of graphs with large chromatic number. XI. Orientations

TL;DR

This paper proves that in graphs with large chromatic number and bounded clique number, any orientation will contain an induced subdigraph isomorphic to certain fixed oriented graphs, confirming conjectures for specific cases.

Contribution

It provides affirmative answers to whether large chromatic number graphs necessarily contain specific oriented subgraphs when edges are oriented, resolving open conjectures for particular cases.

Findings

Confirmed that large chromatic number graphs contain specific oriented subgraphs.

Validated conjectures for the three-edge path and star orientations.

Extended understanding of subgraph structures in graphs with large chromatic number.

Abstract

Fix an oriented graph H, and let G be a graph with bounded clique number and very large chromatic number. If we somehow orient its edges, must there be an induced subdigraph isomorphic to H? Kierstead and Rodl raised this question for two specific kinds of digraph H: the three-edge path, with the first and last edges both directed towards the interior; and stars (with many edges directed out and many directed in). Aboulker et al subsequently conjectured that the answer is affirmative in both cases. We give affirmative answers to both questions.