Rainbow C3's and C4's in edge-colored graphs

Let G(C) 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 d(c) (nu) be the maximum number of distinctly colored edges incident with a vertex nu. We show that if d(c) (nu) > n/2 for every vertex nu is an element of V (G(c)), then G(C) contains at least one rainbow triangle. The bound is sharp. We also obtain a new result about directed C-4's in oriented bipartite graphs and by using it we prove that if H-c is a balanced bipartite graph of order 2n with an edge coloring C such that d(c)(u) > 3n/5 + 1 for every vertex nu is an element of V(HF), then there exists a rainbow C-4 in H-C. (C) 2012 Elsevier B.V. All rights reserved.