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arXiv · 1812.00832

Planar Ramsey graphs

Maria Axenovich·Carsten Thomassen·Ursula Schade·Torsten Ueckerdt
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Abstract

We say that a graph $H$ is planar unavoidable if there is a planar graph $G$ such that any red/blue coloring of the edges of $G$ contains a monochromatic copy of $H$, otherwise we say that $H$ is planar avoidable. I.e., $H$ is planar unavoidable if there is a Ramsey graph for $H$ that is planar. It follows from the Four-Color Theorem and a result of Gon\c{c}alves that if a graph is planar unavoidable then it is bipartite and outerplanar. We prove that the cycle on $4$ vertices and any path are planar unavoidable. In addition, we prove that all trees of radius at most $2$ are planar unavoidable and there are trees of radius $3$ that are planar avoidable. We also address the planar unavoidable notion in more than two colors.

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Maria Axenovich, Carsten Thomassen, Ursula Schade, Torsten Ueckerdt. 2018-12-03. Planar Ramsey graphs. https://arxiv.org/abs/1812.00832

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