Extremal problems for forbidden pairs that imply hamiltonicity

Abstract

Let C denote the claw 1<1,3 , N the net (a graph obtained from a J<3 by attaching a disjoint edge to each vertex of the 1<3 ), W the wounded (a graph obtained from a /{3 by attaching an edge to one vertex and a disjoint path P3 to a second vertex), and Zi the graph consisting of a 1<3 with a path of length i attached to one vertex. For k a fixed ' positive integer and n a sufficiently large integer, the minimal number of edges and the smallest clique in a k-connected graph G of order n that is CY -free (does not contain an induced copy of C or of Y) will be determined for Y a connected subgraph of either P6, N, W, or Z3 . It should be noted that the pairs of graphs CY are precisely those forbidden pairs that imply that any 2-connected graph of order at least 10 is hamiltonian. These extremal numbers give one measure of the relative strengths of the forbidden subgraph conditions that imply a graph is hamiltonian.