Abstract
We partition the set of unlabelled cubic graphs into two disjoint sets, namely “genes” and “descendants”, where the distinction lies in the absence or presence, respectively, of special edge cutsets. We introduce three special operations called breeding operations which accept, as input, two graphs, and output a new graph. The new graph inherits most of the structure of both the input graphs, and so we refer to the input graphs as parents and the output graph as a child. We also introduce three more operations called parthenogenic operations which accept a single descendant as input, and output a slightly more complicated descendant. We prove that every descendant can be constructed from a family of genes via the use of our six operations, and state the result (to be proved in Chap. 3) that this family is unique for any given descendant.
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Notes
- 1.
The name stems from Sir William Hamilton’s investigations of such cycles on the dodecahedron graph around 1856 but Leonhard Euler studied the famous “knight’s tour” on a chessboard in this context as early as 1759.
- 2.
For a more complete study of the prevalence of cubic bridge graphs relative to the total set of cubic non-Hamiltonian graphs, see Filar et al. [9].
References
Balaban, A.T., Harary, F.: The characteristic polynomial does not uniquely determine the topology of a molecule. J. Chem. Doc. 11, 258–259 (1971)
Barnette, D.: Recent progress in combinatorics. In: Tutte, W.T. (ed.) Proceedings of the Third Waterloo Conference on Combinatorics, May 1968. Academic, New York (1969)
Batagelj, V.: Inductive classes of graphs. In: Proceedings of the Sixth Yugoslav Seminar on Graph Theory, Dubrovnik, 1985, pp. 43–56 (1986)
Blanus̆a, D.: Problem cetiriju boja. Glasnik Mat. Fiz. Astr. Ser. II 1, 31–42 (1946)
Brinkmann, G.: Fast generation of cubic graphs. J. Graph Theory 23, 139–149 (1996)
Brinkmann, G., Goedgebeur, J., McKay, B.D.: Generation of cubic graphs. Discret. Math. Theor. Comput. Sci. 13(2), 69–80 (2011)
Ding, G., Kanno, J.: Splitter theorems from cubic graphs. Comb. Probab. Comput. 15, 355–375 (2006)
Eppstein, D.: The traveling salesman problem for cubic graphs. In: Dehne, F., Sack, J-R., Smid, M. (eds.) Algorithms and Data Structures. Volume 2748 of Lecture Notes in Computer Science, pp. 307–318. Springer, Heidelberg (2003)
Filar, J.A., Haythorpe, M., Nguyen, G.T.: A conjecture on the prevalence of cubic bridge graphs. Discuss. Math. Graph Theory 30(1), 175–179 (2010)
Guy, R.K., Harary, F.: On the Möbius ladders. Can. Math. Bull. 10, 493–496 (1967)
Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)
Holton, D.A., Sheehan, J.: The Petersen Graph. Cambridge University Press, Cambridge (1993)
Isaacs, R.: Infinite Families of Nontrivial Trivalent Graphs Which Are Not Tait Colorable. Am. Math. Mon. 82, 221–239 (1975)
Lou, D., Teng, L., Wu, X.: A polynomial algorithm for cyclic edge connectivity of cubic graphs. Australas. J. Comb. 24, 247–259 (2001)
McKay, B.D., Royle, G.F.: Constructing the cubic graphs on up to 20 vertices. Ars Comb. 12A, 129–140 (1986)
Meringer, M.: Fast generation of regular graphs and construction of cages. J. Graph Theory 30, 137–146 (1999)
Ore, O.: The Four-Colour Problem. Academic, New York (1967)
Read, R.C., Wilson, R.J.: An Atlas of Graphs, p. 263. Oxford University Press, Oxford (1998)
Royle, G.: Constructive enumeration of graphs. Ph.D. thesis, University of Western Australia (1987)
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Baniasadi, P., Ejov, V., Filar, J.A., Haythorpe, M. (2016). Genetic Theory for Cubic Graphs. In: Genetic Theory for Cubic Graphs. SpringerBriefs in Operations Research. Springer, Cham. https://doi.org/10.1007/978-3-319-19680-0_1
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