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arXiv · 0803.1637

Large induced trees in K_r-free graphs

Jacob Fox·Po-Shen Loh·Benny Sudakov
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Abstract

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. In this paper, we study the problem of bounding t(G) for graphs which do not contain a complete graph K_r on r vertices. This problem was posed twenty years ago by Erdos, Saks, and Sos. Substantially improving earlier results of various researchers, we prove that every connected triangle-free graph on n vertices contains an induced tree of order \sqrt{n}. When r >= 4, we also show that t(G) >= (\log n)/(4 \log r) for every connected K_r-free graph G of order n. Both of these bounds are tight up to small multiplicative constants, and the first one disproves a recent conjecture of Matousek and Samal.

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Jacob Fox, Po-Shen Loh, Benny Sudakov. 2008-10-25. Large induced trees in K_r-free graphs. https://arxiv.org/abs/0803.1637

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