Short cycle covers on cubic graphs using chosen 2-factor

Abstract

We show that every bridgeless cubic graph $G$ with $m$ edges has a cycle cover of length at most $1.6 m$. Moreover, if $G$ does not contain any intersecting circuits of length $5$, then $G$ has a cycle cover of length $212/135 \cdot m \approx 1.570 m$ and if $G$ contains no $5$-circuits, then it has a cycle cover of length at most $14/9 \cdot m \approx 1.556 m$. To prove our results, we show that each $2$-edge-connected cubic graph $G$ on $n$ vertices has a $2$-factor containing at most $n/10+f(G)$ circuits of length $5$, where the value of $f(G)$ only depends on the presence of several subgraphs arising from the Petersen graph. As a corollary we get that each $3$-edge-connected cubic graph on $n$ vertices has a $2$-factor containing at most $n/9$ circuits of length $5$ and each $4$-edge-connected cubic graph on $n$ vertices has a $2$-factor containing at most $n/10$ circuits of length $5$.