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arXiv · 2104.07927

Polynomial bounds for chromatic number. I. Excluding a biclique and an induced tree

Alex Scott·Paul Seymour·Sophie Spirkl
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Abstract

Let H be a tree. It was proved by Rodl that graphs that do not contain H as an induced subgraph, and do not contain the complete bipartite graph $K_{t,t}$ as a subgraph, have bounded chromatic number. Kierstead and Penrice strengthened this, showing that such graphs have bounded degeneracy. Here we give a further strengthening, proving that for every tree H, the degeneracy is at most polynomial in t. This answers a question of Bonamy, Pilipczuk, Rzazewski, Thomasse and Walczak.

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Alex Scott, Paul Seymour, Sophie Spirkl. 2021-07-25. Polynomial bounds for chromatic number. I. Excluding a biclique and an induced tree. https://arxiv.org/abs/2104.07927

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