Counting rainbow triangles in edge-colored graphs

Abstract

Let $G$ be an edge-colored graph on $n$ vertices. The minimum color degree of $G$, denoted by ${\delta }^c\left(G\right)$, 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 ${\delta }^c\left(G\right)\ge \frac{n+1}{2}$. 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 $\frac{1}{6}{\delta }^c\left(G\right)\left(2{\delta }^c\left(G\right)-n\right)n$, which is best possible by considering the rainbow $k$-partite Turán graph, where its order is divisible by $k$. This means that there are $\Omega \left(n^2\right)$ rainbow triangles in $G$ if ${\delta }^c\left(G\right)\ge \frac{n+1}{2}$, and $\Omega \left(n^3\right)$ rainbow triangles in $G$ if ${\delta }^c\left(G\right)\ge cn$ when $c>\frac{1}{2}$. 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 $F_k$ (i.e., $k$ rainbow triangles sharing a common vertex).