Shortest cycle covers and cycle double covers with large 2-regular subgraphs

TL;DR

This paper investigates the length of shortest cycle covers in snarks and related graphs, providing bounds and confirming conjectures for specific classes, and exploring connections with other graph theory conjectures.

Contribution

The paper proves that many snarks have shortest cycle covers close to 4/3 times the number of edges, and establishes new results for graphs with certain properties.

Findings

Snarks have shortest cycle covers of length approximately 4/3 m.

Graphs with perfect matching index ≤ 4 have cycle covers of length 4/3 m.

Graphs with large circumference have cycle covers close to 4/3 m.

Abstract

In this paper we show that many snarks have shortest cycle covers of length $\frac{4}{3}m+c$ for a constant $c$, where $m$ is the number of edges in the graph, in agreement with the conjecture that all snarks have shortest cycle covers of length $\frac{4}{3}m+o(m)$. In particular we prove that graphs with perfect matching index at most 4 have cycle covers of length $\frac{4}{3}m$ and satisfy the $(1,2)$-covering conjecture of Zhang, and that graphs with large circumference have cycle covers of length close to $\frac{4}{3}m$. We also prove some results for graphs with low oddness and discuss the connection with Jaeger's Petersen colouring conjecture.