Polynomial bounds for chromatic number. III. Excluding a double star

A “double star” is a tree with two internal vertices. It is known that the Gyárfás-Sumner conjecture holds for double stars, that is, for every double star H, there is a function fH such that if G does not contain H as an induced subgraph then χ(G) ≤ fH(ω(G)) (where χ, ω are the chromatic number and the clique number of G). Here we prove that fH can be chosen to be a polynomial.

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