Bo Ning

Spectral radius and Hamiltonian properties of graphs

Let be a graph with minimum degree . The spectral radius of , denoted by , is the largest eigenvalue of the adjacency matrix of . In this note, we mainly prove the following two results.(1) Let be a graph on vertices with . If , then contains a Hamilton path unless .(2) Let be a graph on vertices with . If , then contains a Hamilton cycle unless . As corollaries of our first result, two previous theorems due to Fiedler and Nikiforov and Lu et al. are obtained, respectively. Our second result refines another previous theorem of Fiedler and Nikiforov.

Rainbow triangles in edge-colored graphs

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 [H. Li and G. Wang, Color degree and heterochromatic cycles in edge-colored graphs, European J. Combin. 33 (2012) 1958-1964] is confirmed.

Ore- and Fan-type heavy subgraphs for Hamiltonicity of 2-connected graphs ∗

Abstract 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 ) , d H ( u , v ) = 2 implies that max { d ( u ) , d ( v ) } ≥ n / 2 . We say that G is H - o -heavy ( H - f -heavy) if every induced subgraph of G isomorphic to H is o -heavy ( f -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 S -free) graph is Hamiltonian. Our results extend several previous theorems on forbidden subgraph conditions and heavy subgraph conditions for Hamiltonicity of 2-connected graphs.

Characterizing Heavy Subgraph Pairs for Pancyclicity

Earlier results originating from Bedrossian’s PhD Thesis focus on characterizing pairs of forbidden subgraphs that imply hamiltonian properties. Instead of forbidding certain induced subgraphs, here we relax the requirements by imposing Ore-type degree conditions on the induced subgraphs. In particular, adopting the terminology introduced by Čada, for a graph

Coloring graphs with two odd cycle lengths

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On sufficient conditions for rainbow cycles in edge-colored graphs

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Heavy subgraphs for Hamiltonicity of 2-connected graphs

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