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

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 $7m\mathrm{/}5=1.400m$. We show that every cubic bridgeless graph has a cycle cover of total length at most $34m\mathrm{/}21\approx 1.619m$ and every bridgeless graph with minimum degree three has a cycle cover of total length at most $44m\mathrm{/}27\approx 1.630m$.