3 research outputs found
Planar Ramsey graphs
We say that a graph is planar unavoidable if there is a planar graph
such that any red/blue coloring of the edges of contains a monochromatic
copy of , otherwise we say that is planar avoidable. I.e., is planar
unavoidable if there is a Ramsey graph for 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
vertices and any path are planar unavoidable. In addition, we prove that
all trees of radius at most are planar unavoidable and there are trees of
radius that are planar avoidable. We also address the planar unavoidable
notion in more than two colors