On sufficient conditions for rainbow cycles in edge-colored graphs

Abstract

Let G be an edge-colored graph. We use e(G) and c(G) to denote the number of edges of G and the number of colors appearing on E(G), respectively. For a vertex v∈V(G), the color neighborhood of v is defined as the set of colors assigned to the edges incident to v. A subgraph of G is rainbow if all of its edges are assigned with distinct colors. The well-known Mantel’s theorem states that a graph G on n vertices contains a triangle if e(G)≥⌊n24⌋+1. Rademacher (1941) [8] showed that G contains at least ⌊n2⌋ triangles under the same condition. Li et al. (2014) proved a rainbow version of Mantel’s theorem: An edge-colored graph G has a rainbow triangle if e(G)+c(G)≥n(n+1)∕2. In this paper, we first characterize all graphs G satisfying e(G)+c(G)≥n(n+1)∕2−1 but containing no rainbow triangles. Motivated by Rademacher’s theorem, we then characterize all graphs G which satisfy e(G)+c(G)≥n(n+1)∕2 but contain only one rainbow triangle. We further obtain two results on color neighborhood conditions for the existence of rainbow short cycles. Our results improve a previous theorem due to Broersma et al. (2005). Moreover, we provide a sufficient condition in terms of color neighborhood for the existence of a specified number of vertex-disjoint rainbow cycles.