Colouring graphs with no induced six-vertex path or diamond

TL;DR

This paper proves that the chromatic number of ($P_6$, diamond)-free graphs is bounded by the maximum of 6 and the clique number, and provides a polynomial-time coloring algorithm based on this result.

Contribution

It establishes a chromatic bound for ($P_6$, diamond)-free graphs and develops a polynomial-time coloring algorithm leveraging the Strong Perfect Graph Theorem.

Findings

Chromatic number of ($P_6$, diamond)-free graphs is at most max(6, clique number).

Identifies a unique 6-vertex-critical ($P_6$, diamond, $K_6$)-free graph.

Provides a polynomial-time algorithm to compute the chromatic number for these graphs.

Abstract

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is ($P_6$, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a ($P_6$, diamond)-free graph $G$ is no larger than the maximum of 6 and the clique number of $G$. We do this by reducing the problem to imperfect ($P_6$, diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each case. We also show that there is exactly one 6-vertex-critical ($P_6$, diamond, $K_6$)-free graph. Together with the Lov\'asz theta function, this gives a polynomial time algorithm to compute the chromatic number of ($P_6$, diamond)-free graphs.