Rainbow triangles and cliques in edge-colored graphs

Abstract

For an edge-colored graph, a subgraph is called rainbow if all its edges have distinct colors. We show that if G is an edge-colored graph of order n and size m using c colors on its edges, and m+c≥n+12+k−1 for a non-negative integer k, then G contains at least k rainbow triangles. For n≥3k, we show that this result is best possible, and we completely characterize the class of edge-colored graphs for which this result is sharp. Furthermore, we show that an edge-colored graph G contains at least k rainbow triangles if ∑v∈V(G)dGc(v)≥n+12+k−1 where dGc(v) denotes the number of distinct colors incident to a vertex v.Finally we characterize the edge-colored graphs without a rainbow clique of size at least six that maximizes the sum of edges and colors m+c.Our results answer two questions of Fujita et al., 2019.