Rainbow triangles and cliques in edge-colored graphs

TL;DR

This paper establishes new minimum conditions involving edges and colors in edge-colored graphs that guarantee the existence of a specified number of rainbow triangles, providing tight bounds and characterizations.

Contribution

It introduces sharp bounds and characterizations for rainbow triangles in edge-colored graphs based on edges, colors, and vertex color degrees, answering open questions.

Findings

Minimum conditions for rainbow triangles based on edges and colors.

Characterization of extremal graphs for rainbow triangles.

Results are tight and optimal for large 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]