A counterexample to a conjecture about triangle-free induced subgraphs of graphs with large chromatic number
Abstract
We prove that for every n, there is a graph G with (G) ≥ n and ω(G) ≤ 3 such that every induced subgraph H of G with ω(H) ≤ 2 satisfies (H) ≤ 4. This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5.
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