Hamiltonicity in 3-connected claw-free graphs

Kuipers and Veldman conjectured that any 3-connected claw-free graph with order ν and minimum degree δ ⩾ ( ν + 6 ) 10 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 δ ( H ) ⩾ ( ν + 5 ) 10 , then either H is Hamiltonian, or δ ( H ) = ( ν + 5 ) 10 and the Ryjáčekʹs closure cl ( H ) of H is the line graph of a graph obtained from the Petersen graph P 10 by adding ( ν − 15 ) 10 pendant edges at each vertex of P 10 .