A note on K4-closures in Hamiltonian graph theory

Let G=(V, E) be a 2-connected graph. We call two vertices u and v of G a K4-pair if u and v are the vertices of degree two of an induced subgraph of G which is isomorphic to K4 minus an edge. Let x and y be the common neighbors of a K4-pair u, v in an induced K4−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 K4-pair. Based on these results we define socalled K4-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 K4-closures

Subgraphs, Closures and Hamiltonicity

Closure theorems in hamiltonian graph theory are of the following type: Let G be a 2- connected graph and let u, v be two distinct nonadjacent vertices of G. If condition c(u,v) holds, then G is hamiltonian if and only if G + uv is hamiltonian. We discuss several results of this type in which u and v are vertices of a subgraph H of G on four vertices and c(u, v) is a condition on the neighborhoods of the vertices of H (in G). We also discuss corresponding sufficient conditions for hamiltonicity of G