Abstract
A hole is an induced cycle of length at least 4, and an even-hole is a hole of even length. A cap is a graph consisting of a hole and an additional vertex which is adjacent to exactly two adjacent vertices of the hole. Cameron et al. [3] proved that every (cap, even-hole)-free graph G has χ(G)⩽⌊32ω(G)⌋, and they also proposed a question stating that if χ(G)⩽⌈54ω(G)⌉ for all (cap, even-hole)-free graphs. Wu and Xu [20] showed that every (cap, even-hole)-free graph G has χ(G)⩽⌈43ω(G)⌉. In this paper, we improve this upper bound and show that every (cap, even-hole)-free graph has χ(G)⩽⌈97ω(G)⌉+1.
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