Abstract:Let $G$ be an edge-colored graph on $n$ vertices, and let $\deltac(G)$ denote its minimum color degree. Li and, independently Li, Ning, Xu, and Zhang, proved that every edge-colored graph on $n$ vertices with $\deltac(G) \ge \frac{n+1}{2}$ contains a rainbow triangle. Let $\rt(G)$ denote the number of rainbow triangles in $G$, and define \[ f(n) = \min\{ \rt(G) : |V(G)| = n,\ \deltac(G) \ge (n+1)/2 \}. \] In \cite{LiNingShiZhang2024}, the following open problem was posed: determine all the values of $f(n)$. In this paper, we determine $f(n)$ completely: $f(n) = (n^2-1)/8$ for odd $n\geq 3$, $f(n) = \frac{n^2}{4} - 1$ for all even $n \ge 6,$ and $f(4) = 4$. This resolves an open problem raised in \cite{LiNingShiZhang2024}.
From: Bo Ning [view email]
[v1]
Sat, 20 Jun 2026 15:40:42 UTC (11 KB)
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