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 G is odd 2–factored if for each 2–factor F of G each cycle of F is odd. In this paper, we present a method for constructing odd 2–factored snarks. In particular, we construct two new odd 2–factored snarks that disprove a conjecture by some of the authors. 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.