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

A cycle C in a graph is called stable if there exists no other cycle D in the same graph such that V ( C ) ⊆ V ( D ) . In this paper, we study stable cycles in snarks and we show that if a cubic graph G has a cycle of length at least | 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.