Abstract
In a new class of graphs strictly containing the class of almost claw-free graphs, the class of quasi claw-free graphs and thus the class of claw-free graphs, we show that if the graph G is 2-connected of order n and if the degree sum of any three independent vertices is at least n − 2, then G is hamiltonian. This problem was posed for almost claw-free graphs by Broersma et al. (1996) and was settled by Li and Tian when n ⩾ 79 for another class containing the almost claw-free graphs. We also consider properties of matchings and toughness in this new class. In the main proof, we introduce a technique of global insertion which is a more powerful tool than the usual insertion.
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