Abstract
Let G be a 2-connected claw-free graph on n vertices. Let σ k ( G) be the minimum degree sum among k-element independent set of vertices in G. It is proved that if σ 4( G)⩾ n+3 then G is hamiltonian or else G belong to the known family of graphs. This is a generalization of the best known sufficient condition on hamiltonicity in claw-free 2-connected graphs given independently by Liu, Zhang and Broersma. Moreover, it is shown that the problem HAMILTONIAN CYCLE restricted to claw-free graphs G=( V, E) with σ 3(G)⩾⌊ 3 4 (|G|+3)⌋ has polynomial time complexity. This contrasts sharply with known results on NP-completeness among dense graphs.
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