A Note on Planar Ramsey Numbers for a Triangle Versus Wheels

We assume that the reader is familiar with standard graph-theoretic terminology and refer the readers to Bondy and Murty (2008) for any concept and notation that is not defined here. In this paper, we consider simple, undirected graphs. Given two graphsG andH , the Ramsey numberR(G,H) is the smallest integer n such that every graph F on n vertices contains a copy of G, or its complement F contains a copy of H . The determination of Ramsey numbers is notoriously difficult in general. A variant considered here is the concept of planar Ramsey numbers, introduced by Walker (1967) and rediscovered by Steinberg and Tovey (1993). For two given graphs G and H the planar Ramsey number PR(G,H) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G, or its complement contains a copy of H . It is easy to see that PR(G,H) ≤ R(G,H). Moreover, since the complement of a planar graph F may not be planar, the planar Ramsey number is not symmetric with respect to G and H . The planar Ramsey numbers for all pairs of complete graphs was determined in Steinberg and Tovey (1993). Meanwhile, planar Ramsey numbers for several other pairs of graphs were determined, see e.g. Dudek and Ruciński (2005) and Gorgol and Ruciński (2008). Let G be a graph and G the complement of G. Let U ⊆ V (G), denote by G[U ] the subgraph induced by U in G. We call U a cut set of a connected graph G if G − U is not connected. Let v be a vertex in