A Ramsey property of random regular and k‐out graphs

Abstract

AbstractIn this study we consider a Ramsey property of random ‐regular graphs, . Let be fixed. Then w.h.p. the edges of can be colored such that every monochromatic component has order . On the other hand, there exists a constant such that w.h.p., every ‐coloring of the edges of must contain a monochromatic cycle of length at least . We prove an analogous result for random ‐out graphs.