Hamiltonian cycles in cubic Cayley graphs: the {2,4k,3} case

  • Authors:

  • Henry H. Glover;Klavdija Kutnar;Dragan Marušič

  • Affiliations:

  • Department of Mathematics, Ohio State University, Columbus, USA;University of Primorska, FAMNIT, Koper, Slovenia 6000;University of Ljubljana, IMFM, Ljubljana, Slovenia 1000 and University of Primorska, FAMNIT, Koper, Slovenia 6000

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 2009

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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|a2=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.