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Degree Conditions Restricted to Induced Paths for Hamiltonicity of Claw-heavy Graphs

Acta Mathematica Sinica,English Series 2017年第2期33卷 301-310页

作者： Bin Long LI Bo NING Sheng Gui ZHANG Department of Applied Mathematics Northwestern Polytechnical University Xi'an 710072 P. R. China European Centre of Excellence NTIS 30614 Pilsen Czech Republic Center for Applied Mathematics Tianjin University Tianjin 300072 P. R. China

Broersma and Veldman proved that every 2-connected claw-free and P6-free graph is hamil- tonian. Chen et al. extended this result by proving every 2-connected claw-heavy and P6-free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and P6-o-heavy but not hamiltonian. In this paper, we further give some Ore-type degree conditions restricting to induced copies of P6 of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.

关键词： Hamiltonian graph forbidden subgraph condition degree condition claw-heavy graph,closure theory

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Induced Subgraphs with Large Degrees at End-vertices for Hamiltonicity of Claw-free Graphs

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Acta Mathematica Sinica,English Series 2016年第7期32卷 845-855页

作者： Roman CADA Bin Long LI Bo NING Sheng Gui ZHANG Department of Mathematics University of West Bohemia Centre of Excellence ITI - Institute for Theoretical Computer Science Charles University European Centre of Excellence NTIS - New Technologies for the Information Society P.O. Box 314 306 14 Pilsen Czech Republic Department of Applied Mathematics Northwestern Polytechnical University Xi'an 710072 P. R. China Center for Applied Mathematics Tianjin University Tianjin 300072 P. R. China

A graph is called claw-free if it contains no induced subgrapn lsomorpmc to K1,3. Matthews and Sumner proved that a 2-connected claw-free graph G is Hamiltonian if every vertex of it has degree at least （[V（G）I - 2）/3. At the workshop CSzC （Novy Smokovec, 1993）, Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of N （the graph obtained from a triangle by adding three disjoint pendant edges）. Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of induced copies of Z1 （the graph obtained from a triangle by adding one pendant edge）. Our main result in this paper is a characterization of all graphs H such that a 2-connected claw-free graph G is Hamiltonian if eachend-vertex of every induced copy of H in G has degree at least IV（G）I/3 ＋ 1. This gives an affirmative solution of the conjecture of Broersma up to an additive ***-vertex of every induced copy of H in G has degree at least IV（G）I/3 ＋ 1. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.

关键词： Induced subgraph large degree end-vertex claw-free graph Hamiltonian graph

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