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arXiv · 1903.09725

Large homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs

Maria Axenovich·Casey Tompkins·Lea Weber
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Abstract

For a bipartite graph G, let h(G) be the largest t such that either G or the bipartite complement of G contain K_{t,t}. For a class F of graphs, let h(F)= min {h(G): G\in F}. We say that a bipartite graph H is strongly acyclic if neither H nor its bipartite complement contain a cycle. By Forb(n, H) we denote a set of bipartite graphs with parts of sizes n each, that do not contain H as an induced bipartite subgraph respecting the sides. One can easily show that h(Forb(n,H))= O(n^{1-s}) for a positive s if H is not strongly acyclic. Here, we prove that h(Forb(n, H)) is linear in n for all strongly acyclic graphs except for four graphs.

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Maria Axenovich, Casey Tompkins, Lea Weber. 2019-03-22. Large homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs. https://arxiv.org/abs/1903.09725

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