Edge-disjoint rainbow triangles in edge-colored graphs

Abstract

Let G be an edge-colored graph. A triangle of G is called rainbow if any two edges of the triangle have distinct colors. We use m ( G ) and c ( G ) to denote the number of edges of G and the number of colors appearing on the edges of G , respectively. Li et al. in 2014 proved that every edge-colored graph of order n with m ( G ) + c ( G ) ≥ n ( n + 1 ) / 2 contains a rainbow triangle and this result is best possible. In 2019, Fujita et al. characterized all graphs G satisfying m ( G ) + c ( G ) ≥ n ( n + 1 ) / 2 − 1 but containing no rainbow triangle. In this paper, we conjecture that every edge-colored graph of order n with m ( G ) + c ( G ) ≥ n ( n + 1 ) / 2 + 3 ( k − 1 ) contains k edge-disjoint rainbow triangles. We show that the conjecture holds for k = 2 and 3 and these results are best possible. Furthermore, we characterize all graphs G satisfying m ( G ) + c ( G ) ≥ n ( n + 1 ) / 2 + 2 but not containing two edge-disjoint rainbow triangles. At the end, we propose a conjecture on the number of vertex-disjoint rainbow triangles in an edge-colored graph.