Let $G$ be an edge-colored graph. The color degree of a vertex $v$ of $G$, is
defined as the number of colors of the edges incident to $v$. The color number
of $G$ is defined as the number of colors of the edges in $G$. A rainbow
triangle is one in which every pair of edges have distinct colors. In this
paper we give some sufficient conditions for the existence of rainbow triangles
in edge-colored graphs in terms of color degree, color number and edge number.
As a corollary, a conjecture proposed by Li and Wang (Color degree and
heterochromatic cycles in edge-colored graphs, European J. Combin. 33 (2012)
1958--1964) is confirmed.Comment: Title slightly changed. 13 pages, to appear in European J. Combi
Let G be a graph on n vertices. An induced subgraph H of G is called heavy if there exist two nonadjacent vertices in H with degree sum at least n in G. We say thatIn this paper we characterize all connected graphs R and S other than P 3 (the path on three vertices) such that every 2-connected {R, S}-heavy graph is Hamiltonian. This extends several previous results on forbidden subgraph conditions for Hamiltonian graphs.
Bedrossian characterized all pairs of forbidden subgraphs for a 2-connected graph to be Hamiltonian. Instead of forbidding some induced subgraphs, we relax the conditions for graphs to be Hamiltonian by restricting Ore-and Fan-type degree conditions on these induced subgraphs. Let G be a graph on n vertices and H be an induced subgraph of G. H is called o-heavy if there are two nonadjacent vertices in H with degree sum at least n, and is called f -heavy if for every two vertices u, v ∈ V (H),heavy). In this paper we characterize all connected graphs R and S other than P 3 such that every 2-connected R-f -heavy and S-f -heavy (R-o-heavy and S-f -heavy, R-f -heavy and Sfree) graph is Hamiltonian. Our results extend several previous theorems on forbidden subgraph conditions and heavy subgraph conditions for Hamiltonicity of 2-connected graphs.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.