Counting rainbow triangles in edge-colored graphs

Let G be an edge-colored graph on n vertices. The minimum color degree of G, denoted by dc (G), is defined as the minimum number of colors assigned to the edges incident to a vertex in G. In 2013, Li proved that an edge-colored graph G on n vertices contains a rainbow triangle if dc (G) = n + 1 2. In this paper, we obtain several estimates on the number of rainbow triangles through one given vertex in G. As a consequence, we prove counting results for rainbow triangles in edge-colored graphs. One main theorem states that the number of rainbow triangles in G is at least 1dc (G)(2dc (G) - n) n 6, which is best possible by considering the rainbow k-partite Turan graph, where its order is divisible by k. This means that there are O(n2) rainbow triangles in G if dc (G) = n + 1 2, and O(n3) rainbow triangles in G if =d G cn()c when c > 12. Both results are tight in the sense of the order of the magnitude. We also prove a counting version of a previous theorem on rainbow triangles under a color neighborhood union condition due to Broersma et al., and an asymptotically tight color degree condition forcing a colored friendship subgraph Fk (i.e., k rainbow triangles sharing a common vertex).