The Completion of a Classification 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. Burkard and Hammer gave a necessary condition for a split graph G with |I|<|K| to be Hamiltonian (J. Comb. Theory, Ser. B 28:245–248, 1980). 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 $uv\not\in E$ , where u∈I and v∈K. N.D. Tan and L.X. Hung have classified maximal nonhamiltonian Burkard–Hammer graphs G with minimum degree δ(G)≥|I|−3. Recently, N.D. Tan and Iamjaroen have classified maximal nonhamiltonian Burkard–Hammer graphs with |I|≠6,7 and δ(G)=|I|−4. In this paper, we complete the classification of maximal nonhamiltonian Burkard–Hammer graphs with δ(G)=|I|−4 by finding all such graphs for the case |I|=6,7.