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

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A CLASSIFICATION FOR MAXIMAL NONHAMILTONIAN BURKARD-HAMMER GRAPHS

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

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. 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.

References

[1] M. Behzad and G. Chartrand, Introduction to the theory of graphs (Allyn and Bacon, Boston, 1971).
[2] R.E. Burkard and P.L. Hammer, A note on hamiltonian split graphs, J. Combin. Theory (B) 28 (1980) 245-248, doi: 10.1016/0095-8956(80)90069-6.
[3] V. Chvátal and P.L. Hammer, Aggregation of inequalities in integer programming, Ann. Discrete Math. 1 (1977) 145-162, doi: 10.1016/S0167-5060(08)70731-3.
[4] S. Földes and P.L. Hammer, Split graphs, in: Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La, 1977) pp. 311-315. Congr. Numer., No. XIX, Utilitas Math., Winnipeg, Man. 1977.
[5] S. Földes and P.L. Hammer, On a class of matroid-producing graphs, in: Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely 1976) Vol. 1, 331-352, Colloq. Math. Soc. Janós Bolyai 18 (North-Holland, Amsterdam-New York, 1978).
[6] P.B. Henderson and Y. Zalcstein, A graph-theoretic characterization of the PVchunk class of synchronizing primitive, SIAM J. Computing 6 (1977) 88-108, doi: 10.1137/0206008.
[7] A.H. Hesham and El.R. Hesham, Task allocation in distributed systems: a split graph model, J. Combin. Math. Combin. Comput. 14 (1993) 15-32.
[8] D. Kratsch, J. Lehel and H. Müller, Toughness, hamiltonicity and split graphs, Discrete Math. 150 (1996) 231-245, doi: 10.1016/0012-365X(95)00190-8.
[9] J. Peemöller, Necessary conditions for hamiltonian split graphs, Discrete Math. 54 (1985) 39-47.
[10] U.N. Peled, Regular Boolean functions and their polytope, Chap VI, Ph. D. Thesis (Univ. of Waterloo, Dept. Combin. and Optimization, 1975).
[11] Ngo Dac Tan and Le Xuan Hung, Hamilton cycles in split graphs with large minimum degree, Discuss. Math. Graph Theory 24 (2004) 23-40, doi: 10.7151/dmgt.1210.
[12] Ngo Dac Tan and Le Xuan Hung, On the Burkard-Hammer condition for hamiltonian split graphs, Discrete Math. 296 (2005) 59-72, doi: 10.1016/j.disc.2005.03.008.
[13] Ngo Dac Tan and C. Iamjaroen, Constructions for nonhamiltonian Burkard-Hammer graphs, in: Combinatorial Geometry and Graph Theory (Proc. of Indonesia-Japan Joint Conf., September 13-16, 2003, Bandung, Indonesia) 185-199, Lecture Notes in Computer Science 3330 (Springer, Berlin Heidelberg, 2005).
[14] Ngo Dac Tan and C. Iamjaroen, A necessary condition for maximal nonhamiltonian Burkard-Hammer graphs, J. Discrete Math. Sciences & Cryptography 9 (2006) 235-252.

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