Abstract
It was proved by Glover and Marušič (J. Eur. Math. Soc. 9:775–787, 2007), that cubic Cayley graphs arising from groups G=〈a,x∣a 2=x s=(ax)3=1,…〉 having a (2,s,3)-presentation, that is, from groups generated by an involution a and an element x of order s such that their product ax has order 3, have a Hamiltonian cycle when |G| (and thus also s) is congruent to 2 modulo 4, and have a Hamiltonian path when |G| is congruent to 0 modulo 4.
In this article the existence of a Hamiltonian cycle is proved when apart from |G| also s is congruent to 0 modulo 4, thus leaving |G| congruent to 0 modulo 4 with s either odd or congruent to 2 modulo 4 as the only remaining cases to be dealt with in order to establish existence of Hamiltonian cycles for this particular class of cubic Cayley graphs.
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Alspach, B.: Hamiltonian cycles in vertex-transitive graphs of order 2p. In: Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, FL, 1979). Congress. Numer., vol. XXIII–XX, pp. 131–139. Winnipeg, Manitoba (1979)
Alspach, B., Zhang, C.Q.: Hamilton cycles in cubic Cayley graphs on dihedral groups. Ars Comb. 28, 101–108 (1989)
Anderson, M.S., Richter, R.B.: Self-dual Cayley maps. Eur. J. Comb. 21, 419–430 (2000)
Biggs, N.: Aspects of symmetry in graphs. In: Algebraic methods in graph theory, vols. I, II, Szeged, 1978. Colloq. Math. Soc. János Bolyai, vol. 25, pp. 27–35. North-Holland, Amsterdam (1981)
Bouwer, I.Z. (ed.): The Foster Census. Winnipeg, Manitoba (1988)
Chen, Y.Q.: On Hamiltonicity of vertex-transitive graphs and digraphs of order p 4. J. Comb. Theory Ser. B 72, 110–121 (1998)
Conder, M.D.E., Dobcsanyi, P.: Trivalent symmetric graphs on up to 768 vertices. J. Comb. Math. Comb. Comput. 40, 41–63 (2002)
Conder, M.D.E., Jajcay, R., Tucker, T.: Regular t-balanced Cayley maps. J. Comb. Theory, Ser. B 97, 453–473 (2007)
Conder, M.D.E., Nedela, R.: Symmetric cubic graphs of small girth. J. Comb. Theory, Ser. B 97, 757–768 (2007)
Conder, M.D.E., Nedela, R.: A more detailed classification of symmetric cubic graphs. Preprint
Conway, J.H.: Talk given at the Second British Combinatorial Conference at Royal Holloway College (1971)
Curran, S., Gallian, J.A.: Hamiltonian cycles and paths in Cayley graphs and digraphs—a survey. Discrete Math. 156, 1–18 (1996)
Durnberger, E.: Connected Cayley graphs of semidirect products of cyclic groups of prime order by Abelian groups are Hamiltonian. Discrete Math. 46, 55–68 (1983)
Dobson, E., Gavlas, H., Morris, J., Witte, D.: Automorphism groups with cyclic commutator subgroup and Hamilton cycles. Discrete Math. 189, 69–78 (1998)
Feng, Y.Q., Nedela, R.: Symmetric cubic graphs of girth at most 7. Acta Univ. M. Belii Math. 13, 33–55 (2006)
Frucht, R.: How to describe a graph. Ann. N.Y. Acad. Sci. 175, 159–167 (1970)
Frucht, R., Graver, J.E., Watkins, M.E.: The groups of the generalized Petersen graphs. Proc. Camb. Philos. Soc. 70, 211–218 (1971)
Glover, H.H., Marušič, D.: Hamiltonicity of cubic Cayley graphs. J. Eur. Math. Soc. 9, 775–787 (2007)
Glover, H.H., Yang, T.Y.: A Hamilton cycle in the Cayley graph of the (2,p,3)-presentation of PSL 2(p). Discrete Math. 160, 149–163 (1996)
Horton, J.D., Bouwer, I.Z.: Symmetric Y-graphs and H-graphs. J. Comb. Theory, Ser. B 53, 114–129 (1991)
Jajcay, R.: Automorphism groups of Cayley maps. J. Comb. Theory, Ser. B 59, 297–310 (1993)
Jajcay, R., Širáň, J.: Skew-morphisms of regular Cayley maps. Discrete Math. 244, 167–179 (2002)
Jones, G.A., Surowski, D.B.: Regular cyclic coverings of the Platonic maps. Eur. J. Comb. 21, 333–345 (2000)
Keating, K., Witte, D.: On Hamilton cycles in Cayley graphs in groups with cyclic commutator subgroup. In: Cycles in Graphs, Burnaby, B.C., 1982. North-Holland Math. Stud., vol. 115, pp. 89–102. North-Holland, Amsterdam (1985)
Kutnar, K., Marušič, D.: Hamiltonicity of vertex-transitive graphs of order 4p. Eur. J. Comb. 29, 423–438 (2008)
Kutnar, K., Marušič, D.: A complete classification of cubic symmetric graphs of girth 6. J. Comb. Theory, Ser. B 99, 162–184 (2009)
Lovász, L.: Combinatorial structures and their applications. In: Proc. Calgary Int. Conf., Calgary, Alberta, 1969. Problem, vol. 11, pp. 243–246. Gordon and Breach, New York (1970)
Marušič, D.: Hamilonian circuits in Cayley graphs. Discrete Math. 46, 49–54 (1983)
Marušič, D.: Vertex transitive graphs and digraphs of order p k. In: Cycles in graphs, Burnaby, B.C., 1982. Ann. Discrete Math., vol. 27, pp. 115–128. North-Holland, Amsterdam (1985)
Marušič, D.: Hamiltonian cycles in vertex symmetric graphs of order 2p 2. Discrete Math. 66, 169–174 (1987)
Marušič, D.: On vertex-transitive graphs of order qp. J. Comb. Math. Comb. Comput. 4, 97–114 (1988)
Marušič, D., Parsons, T.D.: Hamiltonian paths in vertex-symmetric graphs of order 5p. Discrete Math. 42, 227–242 (1982)
Marušič, D., Parsons, T.D.: Hamiltonian paths in vertex-symmetric graphs of order 4p. Discrete Math. 43, 91–96 (1983)
Miklavič, Š., Potočnik, P., Willson, S.: Consistent cycles in graphs and digraphs. Graphs Comb. 23, 205–216 (2007)
Nedela, R., Škoviera, M.: Atoms of cyclic connectivity in cubic graphs. Math. Slovaca 45, 481–499 (1995)
Payan, C., Sakarovitch, M.: Ensembles cycliquement stables et graphes cubiques. Cahiers Centre Études Rech. Opér. 17, 319–343 (1975)
Richter, R.B., Širáň, J., Jajcay, R., Tucker, T.W., Watkins, M.E.: Cayley maps. J. Comb. Theory, Ser. B 95, 189–245 (2005)
Turner, J.: Point-symmetric graphs with a prime number of points. J. Comb. Theory 3, 136–145 (1967)
Witte, D.: On Hamilton cycles in Cayley graphs in groups with cyclic commutator subgroup. Discrete Math. 27, 89–102 (1985)
Witte, D.: Cayley digraphs of prime-power order are Hamiltonian. J. Comb. Theory, Ser. B 40, 107–112 (1986)
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Glover, H.H., Kutnar, K. & Marušič, D. Hamiltonian cycles in cubic Cayley graphs: the 〈2,4k,3〉 case. J Algebr Comb 30, 447–475 (2009). https://doi.org/10.1007/s10801-009-0172-5
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DOI: https://doi.org/10.1007/s10801-009-0172-5