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

We show that every bridgeless cubic graph $G$ on $n$ vertices other than the Petersen graph has a 2-factor with at most $2(n-2)\mathrm{/}15$ circuits of length $5$. An infinite family of graphs attains this bound. We also show that $G$ has a 2-factor with at most $n\mathrm{/}5.8\bar{3}$ odd circuits. This improves the previously known bound of $n\mathrm{/}5.41$ [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] ].