Counting rainbow triangles in edge‐colored graphs

Abstract

Let be an edge‐colored graph on vertices. The minimum color degree of , denoted by , is defined as the minimum number of colors assigned to the edges incident to a vertex in . In 2013, Li proved that an edge‐colored graph on vertices contains a rainbow triangle if . In this paper, we obtain several estimates on the number of rainbow triangles through one given vertex in . 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 is at least , which is best possible by considering the rainbow ‐partite Turán graph, where its order is divisible by . This means that there are rainbow triangles in if , and rainbow triangles in if when . 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 (i.e., rainbow triangles sharing a common vertex).