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) ≥ ⌊ n 2 4 ⌋ + 1. Rademacher (1941) showed that G contains at least ⌊ n 2 ⌋ triangles under the same condition. Li, Ning, Xu and Zhang (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, Li, Woeginger, and Zhang (2005). Moreover, we provide a sufficient condition in terms of color neighborhood for the existence of a specified number of vertex-disjoint rainbow cycles.

Rainbow triangles and cliques in edge-colored graphs

“…We can also characterize the graphs G with m(G) + c(G) ≥ n+1 2 + k − 1 that contain exactly k rainbow triangles. This generalizes the main results of [5], where this is proved for k ∈ {0, 1}.…”

“…Answering another question of [5], we can in fact show that these are essentially all the extremal graphs if k ≥ 6, except when one partite set of T n,k−2 has size 1. To this end, let H k be the set of all edge-colored graphs such that for every graph H ∈ H k on n vertices either (I) H is isomorphic to H n,k−2 or (II) ⌊n/(k − 2)⌋ = 1, H is complete, c(H) = t n,k−2 + 1, and H contains a rainbow T n,k−2 as a subgraph but no rainbow K k .…”

“…− 1 that do not contain a rainbow triangle. Fujita, Ning, Xu, and Zhang [5] characterized all extremal graphs G for Theorem 1 such that m(G) + c(G) is maximum but G contains at most one rainbow triangle.…”

“…This motivates the question in [5] whether one can find a function f (k) such that a graph G on n vertices with m(G) + c(G) ≥ n+1 2 + f (k) necessarily contains k rainbow triangles, which can be seen as a rainbow version of Erdős' conjecture mentioned above. Our first result answers this question.…”

Graph colorings under global structural conditions

“…In general, there are color degree conditions, forbidden subgraph conditions, and so on. There has been a large amount of published literature in this subject; see [1,11,15,22,26,28,29,31,40,41] for examples and the references therein. But, this kind of research is obviously different from the one here.…”