Five‐cycle double cover and shortest cycle cover

Abstract

The 5‐even subgraph cycle double cover conjecture (5‐CDC conjecture) asserts that every bridgeless graph has a 5‐even subgraph double cover. A shortest even subgraph cover of a graph is a family of even subgraphs which cover all the edges of and the sum of their lengths is minimum. It is conjectured that every bridgeless graph has an even subgraph cover with total length at most . In this paper, we study those two conjectures for weak oddness 2 cubic graphs and present a sufficient condition for such graphs to have a 5‐CDC containing a member with many vertices. As a corollary, we show that for every oddness 2 cubic graph satisfying the sufficient condition has a 4‐even subgraph ‐cover with total length at most . We also show that every oddness 2 cubic graph with girth at least 30 has a 5‐CDC containing a member of length at least and thus it has a 4‐even subgraph ‐cover with total length at most .