Note on Rainbow Triangles in Edge-Colored Graphs

Abstract

Let G be a graph with an edge-coloring c, and let $\delta ^c(G)$ denote the minimum color-degree of G. A subgraph of G is called rainbow if any two edges of the subgraph have distinct colors. In this paper, we consider color-degree conditions for the existence of rainbow triangles in edge-colored graphs. At first, we give a new proof for characterizing all extremal graphs G with $\delta ^c(G)\ge \frac{n}{2}$ that do not contain rainbow triangles, a known result due to Li et al. Then, we characterize all complete graphs G without rainbow triangles under the condition $\delta ^c(G)=log_2n$, extending a result due to Li, Fujita and Zhang. Hu, Li and Yang showed that G contains two vertex-disjoint rainbow triangles if $\delta ^c(G)\ge \frac{n+2}{2}$ when $n\ge 20$. We slightly refine their result by showing that the result also holds for $n\ge 6$, filling the gap of n from 6 to 20. Finally, we prove that if $\delta ^c(G)\ge \frac{n+k}{2}$ then every vertex of an edge-colored complete graph G is contained in at least k rainbow triangles, generalizing a result due to Fujita and Magnant. At the end, we mention some open problems.