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. 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.
Highlights
Ramsey’s theorem [16] claims that any graph is Ramsey in the class of all complete graphs, i.e., for any graph G and any number k of colors there is a sufficiently large complete graphA result of Goncalves [8] states that any planar graph can be edge-colored in two colors so that each color class is an outerplanar graph
The Four Color Theorem [2] implies that any planar graph is a union of two bipartite graphs
Any outerplanar graph admits an orientation with out-degree at most 2 at each vertex
Summary
Ramsey’s theorem [16] claims that any graph is Ramsey in the class of all complete graphs, i.e., for any graph G and any number k of colors there is a sufficiently large complete graph. Encoding the colors by bit-vectors of length k, the bipartition classes of the i-th bipartite graph correspond to the bit-vectors with a 0, respectively a 1, in the i-th position. This shows that any planar unavoidable graph is bipartite and outerplanar and gives necessary conditions for planar unavoidability. If H is a cycle on 4 vertices, a tree of radius at most 2, or a generalized broom, H is planar unavoidable. The result shows that odd cycles and non-outerplanar graphs are planar avoidable, and some trees.
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