Geodesic
A classification for maximal nonhamiltonian Burkard-Hammer graphs
Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 1, pp. 67-89.

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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. Recently, Ngo Dac Tan and Le Xuan Hung have classified maximal nonhamiltonian Burkard-Hammer graphs G with minimum degree δ(G) ≥ |I|- 3. In this paper, we classify maximal nonhamiltonian Burkard-Hammer graphs G with |I| ≠ 6,7 and δ(G) = |I| - 4.
Keywords: split graph, Burkard-Hammer condition, Burkard-Hammer graph, hamiltonian graph, maximal nonhamiltonian split graph 
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Tan, Ngo; Iamjaroen, Chawalit. A classification for maximal nonhamiltonian Burkard-Hammer graphs. Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 1, pp. 67-89. http://geodesic.mathdoc.fr/item/DMGT_2008_28_1_a4/

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