Abstract
A benzenoid systemH is a finite connected subgraph of the infinite hexagonal lattice with out cut bonds and non-hexagonal interior faces. The branching graphG ofH consists of all vertices ofH of degree 3 and bonds among them. In this paper, the following results are obtained:
-
(1)
A necessary condition for a benzenoid system to have a Hamiltonian circuit.
-
(2)
A necessary and sufficient condition for a benzenoid system to have a Hamiltonian path.
-
(3)
A characterization of connected subgraphs of the infinite hexagonal lattice which are branching graphs of benzenoid systems.
-
(4)
A proof that if a disconnected subgraph G of the infinite hexagonal lattice given along with the positions of its vertices is the branching graph of a benzenoid system H, then H is unique.
Similar content being viewed by others
References
M.R. Garey and D.S. Johnson,Computers and Intractability, a Guide to the Theory of NPcompleteness (Freeman, New York, 1979).
M.R. Garey, D.S. Johnson and E. Tarjan, SIAM J. Comp. 4 (1976) 704.
W.R. Hamilton, Letter to John T. Graves on the Icosian, 17 Oct., 1856, in:The Mathematical Papers of Sir William Rowan Hamilton, eds. H. Halberstam and R.E. Ingram, Vol. 3 (Algebra) (Cambridge University Press, 1931) pp. 612-25.
G.A. Dirac, Proc. London Math. Soc. 2 (1952) 69.
V. Chvatal, J. Comb. Theor. B12 (1972) 163.
J.A. Bondy, Canad. Math. Bull. 15 (1972) 57.
J.A. Bondy and U.S.R. Murty, Graph Theory and Applications (Elsevier, New York, London, 1976) Ch. 4.
C. Berge,Graphs and Hypergraphs (North-Holland, Amsterdam, 1973) Ch. 10.
G.G. Hall and D.R. Dias, J. Math. Chem. 3 (1989) 233.
E.C. Kirby, J. Math. Chem. 4 (1990) 31.
E.C. Kirby, J. Chem. Soc. Farad. Trans. 86 (1990) 447.
E.C. Kirby, J. Math. Chem. 8 (1991) 77.
E.C. Kirby, J. Math. Chem. 11 (1992) 187.
J.V. Knop, W.R. Miiller, K. Szymanski and N. Trinajstic, J. Comp. Chem. 7 (1986) 547.
I. Gutman and E.C. Kirby, Match 26 (1991) 111–122.
E.C. Kirby and I. Gutman, J. Math. Chem. 13 (1993) 359.
I. Gutman, in:Advances in the Theory of Benzenoid Hydrocarbons II, ed. I. Gutman (Springer, Berlin, 1992) pp. 1–29.
M. Randić, Chem. Phys. Lett. 36 (1974) 68.
N. Trinajstic, S. Nikolić, J.V. Knop, W.R. Müller and K. Szymanski,Computational Chemical Graph Theory (Ellis Horwood, New York, London, 1991).
P. Hansen and M. Zheng, J. Mol. Struct. (Theochem) 257 (1992) 75.
X. Li and F. Zhang, Match 25 (1990)151.
I. Gutman and S.J. Cyvin,Introduction to the Theory of Benzenoid Hydrocarbons (Springer, Berlin, 1989).
I. Gutman, in:Graph Theory, eds. D. Cvetković, I. Gutman, T. Pisanaki and R. Tošć (Univ. Novi Sad, Novi Sad, 1983) pp. 151–160.
M. Randic, J. Chem. Soc. Faraday Trans. 11 (1976) 232–243.
B. Dzonova-Jerman-Blazič and N. Trinajstic, Compt. Chem. 6 (1982) 121.
J.V. Knop, K. Szymanski and N. Trinajstic, Comp. Math. Appl. 10 (1984) 369.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hansen, P., Zheng, M. Hamiltonian circuits, Hamiltonian paths and branching graphs of benzenoid systems. J Math Chem 17, 15–33 (1995). https://doi.org/10.1007/BF01165135
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01165135