Odd 2-factored snarks

  • Authors:

  • M. Abreu;D. Labbate;R. Rizzi;J. Sheehan

  • Affiliations:

  • -;-;-;-

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2014

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Abstract

A snark is a cubic cyclically 4-edge connected graph with edge chromatic number four and girth at least five. We say that a graph G is odd 2-factored if for each 2-factor F of G each cycle of F is odd. Some of the authors conjectured in Abreu et al. (2012) [4] that a snark G is odd 2-factored if and only if G is the Petersen graph, Blanusa 2, or a flower snark J(t), with t=5 and odd. Brinkmann et al. (2013) [10] have obtained two counterexamples that disprove this conjecture by performing an exhaustive computer search of all snarks of order n@?36. In this paper, we present a method for constructing odd 2-factored snarks. In particular, we independently construct the two odd 2-factored snarks that yield counterexamples to the above conjecture. Moreover, we approach the problem of characterizing odd 2-factored snarks furnishing a partial characterization of cyclically 4-edge connected odd 2-factored snarks. Finally, we pose a new conjecture regarding odd 2-factored snarks.