Short Cycle Covers of Cubic Graphs and Graphs with Minimum Degree Three

TL;DR

This paper improves bounds on cycle covers in cubic and minimum degree three bridgeless graphs, moving closer to the conjectured optimal bounds and providing new upper limits for total cycle cover length.

Contribution

It establishes new upper bounds for cycle covers in cubic and minimum degree three bridgeless graphs, advancing understanding of the Shortest Cycle Cover Conjecture.

Findings

Cubic bridgeless graphs have cycle covers with total length at most 34m/21.

Bridgeless graphs with minimum degree three have cycle covers with total length at most 44m/27.

The results improve previous bounds and support the conjecture's validity.

Abstract

The Shortest Cycle Cover Conjecture of Alon and Tarsi asserts that the edges of every bridgeless graph with $m$ edges can be covered by cycles of total length at most $7m/5=1.400m$. We show that every cubic bridgeless graph has a cycle cover of total length at most $34m/21\approx 1.619m$ and every bridgeless graph with minimum degree three has a cycle cover of total length at most $44m/27\approx 1.630m$.