Rainbow [formula omitted]’s and [formula omitted]’s in edge-colored graphs

Abstract

Let Gc be a graph of order n with an edge coloring C. A subgraph F of Gc is rainbow if any pair of edges in F have distinct colors. We introduce examples to show that some classic problems can be transferred into problems on rainbow subgraphs. Let dc(v) be the maximum number of distinctly colored edges incident with a vertex v. We show that if dc(v)>n/2 for every vertex v∈V(Gc), then Gc contains at least one rainbow triangle. The bound is sharp. We also obtain a new result about directed C4’s in oriented bipartite graphs and by using it we prove that if Hc is a balanced bipartite graph of order 2n with an edge coloring C such that dc(u)>3n5+1 for every vertex v∈V(Hc), then there exists a rainbow C4 in Hc.