A necessary condition for maximal nonhamiltonian Burkard-Hammer graphs

Abstract

A graph G=(V, E) is called a split graph if there exists a partition V=I∪K such that the subgraphs G[I] and G[K] of G induced by I and K are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary condition for a split graph G with |I|<|K| to be hamiltonian. We will call a split graph G with |I|<|K| satisfying this condition a Burkard-Hammer graph. Further, a split graph G is called a maximal nonhamiltonian split graph if G is nonhamiltonian but G+uv is hamiltonian for every uv∉E where u∈I and v∈K. In this paper, we prove that a maximal nonhamiltonian Burkard-Hammer graph G with the minimum degree δ(G)=|I|–k, where k≥3, must have |I|≥k+2 and no vertices with exactly k+1, k+2, …, |I|–1 neighbours in I and if k≥3 and |I|>k+2, then G also has no vertices with exactly k neighbours in I. We show further that the above obtained results are best possible.