Coloring ([formula omitted], kite)-free graphs with small cliques

Abstract

Let Pn and Kn denote the induced path and complete graph on n vertices, respectively. The kite is the graph obtained from a P4 by adding a vertex and making it adjacent to all vertices in the P4 except one vertex with degree 1. A graph is (P5, kite)-free if it has no induced subgraph isomorphic to a P5 or a kite. For a graph G, the chromatic number of G (denoted by χ(G)) is the minimum number of colors needed to color the vertices of G such that no two adjacent vertices receive the same color, and the clique number of G is the size of a largest clique in G. Here, we are interested in coloring the class of (P5, kite)-free graphs with small clique number. It is known that every (P5, kite, K3)-free graph G satisfies χ(G)≤3, every (P5, kite, K4)-free graph G satisfies χ(G)≤4, and that every (P5, kite, K5)-free graph G satisfies χ(G)≤6. In this paper, we show the following: •Every (P5, kite, K6)-free graph G satisfies χ(G)≤7.•Every (P5, kite, K7)-free graph G satisfies χ(G)≤9. We also give examples to show that the above bounds are tight. Based on these partial results and some other examples, we conjecture that every (P5, kite)-free graph G satisfies χ(G)≤⌊3ω(G)2⌋, and that the bound is tight.