Every (cap, even hole)-free graph G satisfies χ(G) ≤ ⌈5/4 ω(G)⌉, and every (cap, even hole, 5-hole)-free graph satisfies χ(G) ≤ ⌈7/6 ω(G)⌉, with both bounds attained by clique blowups of odd holes.
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The optimal binding function for (cap, even hole)-free graphs
Every (cap, even hole)-free graph G satisfies χ(G) ≤ ⌈5/4 ω(G)⌉, and every (cap, even hole, 5-hole)-free graph satisfies χ(G) ≤ ⌈7/6 ω(G)⌉, with both bounds attained by clique blowups of odd holes.