Colouring graphs with no induced six-vertex path or diamond

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 Lovász theta function, this gives a polynomial time algorithm to compute the chromatic number of (P 6 , diamond)-free graphs.

The Chromatic Number of -Free Graphs

“…In 2021, Cameron et al [2] improved the χ-bounding function of (P 6 , diamond)-free graphs to ω(G) + 3. In a recent paper [8], Goedgebeur et al proved that every (P 6 , diamond)-free graph G satisfies χ(G) ≤ max{6, ω(G)}.…”