Obstructions for three-coloring and list three-coloring $H$-free graphs

TL;DR

This paper characterizes all graphs H for which there are finitely many minimal non-three-colorable and minimal obstructions for list 3-colorability in H-free graphs, solving a previously open problem.

Contribution

It provides a complete characterization of graphs H with finitely many minimal obstructions for three-coloring and list three-coloring in H-free graphs, extending prior results to disconnected graphs.

Findings

Finitely many minimal non-three-colorable H-free graphs identified for certain H.

Finitely many minimal obstructions for list 3-colorability characterized.

Solved an open problem posed by Golovach et al.

Abstract

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We characterize all graphs $H$ for which there are only finitely many minimal non-three-colorable $H$-free graphs. Such a characterization was previously known only in the case when $H$ is connected. This solves a problem posed by Golovach et al. As a second result, we characterize all graphs $H$ for which there are only finitely many $H$-free minimal obstructions for list 3-colorability.