Irreducible pseudo 2-factor isomorphic cubic bipartite graphs

  • Authors:

  • Marién Abreu;Domenico Labbate;John Sheehan

  • Affiliations:

  • Dipartimento di Matematica e Informatica, Università degli Studi della Basilicata, Potenza, Italy 85100;Dipartimento di Matematica, Politecnico di Bari, Bari, Italy 70125;Department of Mathematical Sciences, King's College, Old Aberdeen, Scotland, UK AB24 3UE

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2012

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Abstract

A bipartite graph is pseudo 2-factor isomorphic if the number of circuits in each 2-factor of the graph is always even or always odd. We proved (Abreu et al., J Comb Theory B 98:432---442, 2008) that the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graph of girth 4 is K 3,3, and conjectured (Abreu et al., 2008, Conjecture 3.6) that the only essentially 4-edge-connected cubic bipartite graphs are K 3,3, the Heawood graph and the Pappus graph. There exists a characterization of symmetric configurations n 3 due to Martinetti (1886) in which all symmetric configurations n 3 can be obtained from an infinite set of so called irreducible configurations (Martinetti, Annali di Matematica Pura ed Applicata II 15:1---26, 1888). The list of irreducible configurations has been completed by Boben (Discret Math 307:331---344, 2007) in terms of their irreducible Levi graphs. In this paper we characterize irreducible pseudo 2-factor isomorphic cubic bipartite graphs proving that the only pseudo 2-factor isomorphic irreducible Levi graphs are the Heawood and Pappus graphs. Moreover, the obtained characterization allows us to partially prove the above Conjecture.