Hamiltonian Claw-Free Graphs and o-Heavy Graphs Involving Induced Cycles

In this paper, we show that every 3-connected claw-free graph such that every induced cycle of length at least 4 has at most 8 edges contained in a triangle is hamiltonian. This implies that every 3-connected (Formula presented.) -free graph is hamiltonian. We also show that every 3-connected o-heavy graph whose induced cycle of length is at most 8 is hamiltonian and that every 2-connected claw-free graph of longest induced cycle with length at least (Formula presented.) is hamiltonian. These results are all best possible. As a byproduct, we show that it is NP-complete to determine the length of a longest induced cycle of a line graph.