A note on chromatic number of (cap, even hole)-free graphs

A hole is an induced cycle of length at least 4, and a cap is obtained from a hole by adding a new vertex and joining it to exactly two adjacent vertices of the hole. Let G be a graph containing neither caps nor holes of even lengths as induced subgraphs. Cameron et al. (2018) showed that chi(G) <= left perpendicular3 omega(G)/2right perpendicular and asked whether chi(G) <= inverted right perpendicular5 omega(G)/4inverted left perpendicular, where chi(G) and omega(G) denote the chromatic number and clique number of G, respectively. In this paper, we show that chi(G) <= inverted right perpendicular4 omega(G)/3inverted left perpendicular, and chi(G) <= inverted right perpendicular5 omega(G)/4inverted left perpendicular if G further has no 5-cycles sharing exactly one edge. This improves the conclusion of Cameron et al. (2018), and conditionally answers their question in affirmative. (C) 2018 Elsevier B.V. All rights reserved.