Avoiding 5-circuits in a 2-factor of cubic graphs

Abstract: We show that every bridgeless cubic graph $G$ on $n$ vertices other than the Petersen graph has a 2-factor with at most $2(n-2)/15$ circuits of length $5$. An infinite family of graphs attains this bound. We also show that $G$ has a 2-factor with at most $n/5.8\overline{3}$ odd circuits. This improves the previously known bound of $n/5.41$ [Luko\v{t}ka, M\'a\v{c}ajov\'a, Maz\'ak, \v{S}koviera: Small snarks with large oddness, arXiv:1212.3641 [cs.DM] ].