Forbidden Subgraphs, Distance,and Hamiltonicity

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Forbidden Subgraphs, Distance,and Hamiltonicity

A graph is claw-free if it contains no induced subgraph isomorphic to a K1,3.This paper studies hamiltonicity in 3-connected claw-free graphs.Four generation of Shepherd's result4] are obtained.For example,we show that if G is.3-connected claw-free graph and(1)if for each vertex V the set of venices at distance three from v doesn'tcontain and independent subset of size three,then G is hamiltonian;(2) if G contains no induced subgraph with degree sequence(1,1,1,2,2,2,3,3,3),so that ear vertel of degree is adjacent to a vertex of degree i + 1 for i=1,2,then G is hamiltonoan. Furthermore,we obtain a generalization of both(1) and(2),in which the graphs F1 and F2coatain an the known forbidded subgraphs given in3] as indeced subgraphs.