Counting rainbow triangles in edge-colored graphs

TL;DR

This work investigates counting rainbow triangles in edge-colored graphs under minimum color-degree constraints. It proves a tight lower bound $rt(G)\ge \frac{1}{6}\delta^c(G)(2\delta^c(G)-n)n$ for graphs with $\delta^c(G)\ge \frac{n+1}{2}$, with equality realized by rainbow $k$-partite Turán graphs, establishing a strong supersaturation phenomenon. The authors also derive refined counts for rainbow triangles through a specified vertex in terms of the vertex's monochromatic degree and color-neighborhood structure, obtain a counting version under a color-neighborhood union condition, and prove an asymptotically tight color-degree condition guaranteeing a colored friendship subgraph $F_k$. The results imply $\Omega(n^2)$ rainbow triangles at the base threshold and $\Omega(n^3)$ when the color degree scales linearly with $n$ (for $c>\tfrac12$), and they connect extremal constructions with rainbow subgraph counts, informing both theory and potential applications in graph coloring problems.

Abstract

Let $G$ be an edge-colored graph on $n$ vertices. The minimum color degree of $G$, denoted by $δ^c(G)$, is defined as the minimum number of colors assigned to the edges incident to a vertex in $G$. In 2013, H. Li proved that an edge-colored graph $G$ on $n$ vertices contains a rainbow triangle if $δ^c(G)\geq \frac{n+1}{2}$. In this paper, we obtain several estimates on the number of rainbow triangles through one given vertex in $G$. As consequences, 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 $\frac{1}{6}δ^c(G)(2δ^c(G)-n)n$, which is best possible by considering the rainbow $k$-partite Turán graph, where its order is divisible by $k$. This means that there are $Ω(n^2)$ rainbow triangles in $G$ if $δ^c(G)\geq \frac{n+1}{2}$, and $Ω(n^3)$ rainbow triangles in $G$ if $δ^c(G)\geq cn$ when $c>\frac{1}{2}$. Both results are tight in 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).

Theorems & Definitions (31)

• Theorem 1: L13
• Theorem 2: LNXZ14
• Remark 1
• Theorem 3
• Theorem 4
• Example 1
• Theorem 5
• Example 2
• Example 3
• Theorem 6