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 $G$ is a family of even subgraphs which cover all the edges of $G$ and the sum of their lengths is minimum. It is conjectured that every bridgeless graph $G$ has an even subgraph cover with total length at most $\frac{21}{15}\mid E\left(G\right)\mid$. 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 $G$ satisfying the sufficient condition has a 4-even subgraph $\left(1,2\right)$-cover with total length at most $\frac{20}{15}\mid E\left(G\right)\mid +2$. We also show that every oddness 2 cubic graph $G$ with girth at least 30 has a 5-CDC containing a member of length at least $\frac{9}{10}\mid V\left(G\right)\mid$ and thus it has a 4-even subgraph $\left(1,2\right)$-cover with total length at most $\frac{21}{15}\mid E\left(G\right)\mid$.