Discussiones Mathematicae Graph Theory 28(1) (2008) 67-89
doi: 10.7151/dmgt.1392

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Ngo Dac Tan

Institute of Mathematics
18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam
e-mail: ndtan@math.ac.vn

Chawalit Iamjaroen

Department of Mathematics, Mahasarakham University
Kamrieng, Kantarawichai, Mahasarakham 44150, Thailand
e-mail: chawalit.i@msu.ac.th


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.

2000 Mathematics Subject Classification: Primary 05C45; Secondary 05C75.


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Received 22 September 2006
Revised 21 May 2007
Accepted 21 May 2007