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 $m+c\geq \binom{n+1}{2}+k-1$ for a non-negative integer $k$, then $G$ contains at least $k$ rainbow triangles. For $n\geq 3k$, 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 $\sum\limits_{v\in V(G)} d^c_G(v)\geq \binom{n+1}{2}+k-1$ where $d_G^c(v)$ 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 maximize the sum of edges and colors $m+c$. Our results answer two questions of Fujita, Ning, Xu and Zhang [On sufficient conditions for rainbow cycles in edge-colored graph, arXiv:1705.03675, 2017]