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

In this paper we show that many snarks have shortest cycle covers of length 43m + 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 43m+ o(m). In particular we prove that graphs with perfect matching index at most 4 have cycle covers of length 4 3m and satisfy the (1, 2)-covering conjecture of Zhang, and that graphs with large circumference have cycle covers of length close to 4 3m. We also prove some results for graphs with low oddness and discuss the connection with Jaeger’s Petersen colouring conjecture.