Hamiltonicity in 3-connected claw-free graphs

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