Odd 2-factored snarks

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 GG is odd 2-factored   if for each 2-factor FF of GG each cycle of FF is odd. Some of the authors conjectured in Abreu et al. (2012)  [4] that a snark GG is odd 2-factored if and only if GG is the Petersen graph, Blanuša 2, or a flower snark J(t)J(t), with t≥5t≥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≤36n≤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.