A note on K4-closures in hamiltonian graph theory

Abstract

Let G=( V, E) be a 2-connected graph. We call two vertices u and v of G a K 4-pair if u and v are the vertices of degree two of an induced subgraph of G which is isomorphic to K 4 minus an edge. Let x and y be the common neighbors of a K 4-pair u, v in an induced K 4− e. We prove the following result: If N(x)⌣N(y)⊆N(u)⌣N(v)⌣{u,v}, then G is hamiltonian if and only if G+ uv is h amiltonian. As a consequence, a claw-free graph G is hamiltonian if and only if G+ uv is hamiltonian, where u, v is a K 4-pair. Based on these results we define socalled K 4-closures of G. We give infinite classes of graphs with small maximum degree and large diameter, and with many vertices of degree two having complete K 4-closures.