Rainbow ’s and ’s in edge-colored graphs

Let G c be a graph of order n with an edge coloring C . A subgraph F of G c 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 ( v ) be the maximum number of distinctly colored edges incident with a vertex v . We show that if d c ( v ) > n / 2 for every vertex v ∈ 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 2 n with an edge coloring C such that d c ( u ) > 3 n 5 + 1 for every vertex v ∈ V ( H c ) , then there exists a rainbow C 4 in H c .