Note on Rainbow Triangles in Edge-Colored Graphs

Let G be a graph with an edge-coloring c , and let δ 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 δ c ( G ) ≥ 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 δ c ( G ) = l o g 2 n , extending a result due to Li, Fujita and Zhang. Hu, Li and Yang showed that G contains two vertex-disjoint rainbow triangles if δ c ( G ) ≥ n + 2 2 when n ≥ 20 . We slightly refine their result by showing that the result also holds for n ≥ 6 , filling the gap of n from 6 to 20. Finally, we prove that if δ c ( G ) ≥ 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.