• DOI:10.37236/8366
• Corpus ID: 54200115

Planar Ramsey graphs

@article{Axenovich2019PlanarRG,
  title={Planar Ramsey graphs},
  author={Maria Axenovich and Carsten Thomassen and Ursula Schade and Torsten Ueckerdt},
  journal={Electron. J. Comb.},
  year={2019},
  volume={26},
  pages={4}
}

• M. Axenovich, C. Thomassen, +1 author T. Ueckerdt
• Published 3 December 2018
• Mathematics
• Electron. J. Comb.

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. That is, $H$ 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 Gonçalves 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.