  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
$44m/27\approx 1.630m$.
