Hamiltonicity in 3-connected claw-free graphs

Kuipers and Veldman conjectured that any 3-connected claw-free graph with order v and minimum degree δ ≥ (v+6)/10 is Hamiltonian for v sufficiently large. In this paper, we prove that if H is a 3-connected claw-free graph with sufficiently large order v, and if δ (H) ≥ (v + 5)/10, then either H is Hamiltonian, or δ(H) = (v+5)/10 and the Ryjacek's closure cl(H) of H is the line graph of a graph obtained from the Petersen graph P10 by adding (v - 15)/10 pendant edges at each vertex of P10.

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