Short cycle covers on cubic graphs using chosen 2-factor

TL;DR

This paper establishes improved bounds on cycle covers in bridgeless cubic graphs, showing that certain connectivity and subgraph conditions lead to shorter cycle covers, with implications for graph theory and combinatorics.

Contribution

It introduces new bounds on cycle cover lengths for bridgeless cubic graphs based on their connectivity and subgraph structure, especially involving 2-factors and circuits of length 5.

Findings

Cycle cover length at most 1.6 times the number of edges in general bridgeless cubic graphs.

Reduced cycle cover length bounds for graphs without intersecting 5-circuits.

Existence of 2-factors with limited 5-circuits depending on graph connectivity.

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 $5$.