On stable cycles and cycle double covers of graphs with large circumference

A cycle CC in a graph is called stable   if there exists no other cycle DD in the same graph such that V(C)⊆V(D)V(C)⊆V(D). In this paper, we study stable cycles in snarks and we show that if a cubic graph GG has a cycle of length at least |V(G)|−9|V(G)|−9 then it has a cycle double cover. We also give a construction for an infinite snark family with stable cycles of constant length and answer a question by Kochol by giving examples of cyclically 5-edge connected snarks with stable cycles.