Barrett McCormick Marquette University

Indianapolis, IN

For a family of k-connected Hamiltonian claw-free graphs H of sufficiently large order n with δ(H) ≥ 3 and t vertices, we show that for a given number ε and given integers p ≥ t > 0, H has a finite obstruction set in which each member is a k-edge-connected K_3-free graph of order at most max{p/t + 2t, 3p/t +2t - 7} and without spanning closed trails. We found the best possible values of p and ε for some t ≥ 2 when the obstruction set is empty or has the Petersen graph only. In particular, we prove the following for such graphs H: (a) For k = 2 and a given t (1 ≤ t ≤ 4), if the degree condition is δ_t(H) ≥ (n + 1)/3 and δ(H) ≥ 3, then H is Hamiltonian; (b) For k = 3 and t = 2, (i) if the degree condition is δ_2(H) ≥ (n + 12)/9, then H is Hamiltonian and (ii) if δ_2(H) ≥ (n + 9)/10, then either H is Hamiltonian or H can be characterized by the Petersen graph; (c) For k = 3 and t = 3, (i) if δ_3(H) ≥ (n + 9)/8, then H is Hamiltonian and if (ii) δ_3(H) ≥ (n + 6)/9, then either H is Hamiltonian, or H can be characterized by the Petersen graph.