The (5, 2)-cycle-cover conjecture

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. $(5,2)$-cycle-cover conjecture

   Let $G$ be a bridgeless graph, not necessarily cubic. An even subgraph is a subgraph in which every vertex of $G$ has even degree. $(5,2)$-cycle-cover conjecture. The graph $G$ contains five even subgraphs such that every edge belongs to exactly two of them. The conjecture is presented as a classical consequence implied by the Petersen coloring conjecture.

   source: Vahan V. Mkrtchyan, “A remark on Petersen coloring conjecture of Jaeger”, arXiv:1201.4472 (2012).

2. The $(5,2)$-cycle-cover conjecture

   Let $G$ be a bridgeless graph, not necessarily cubic. An even subgraph is a subgraph in which every vertex has even degree. The $(5,2)$-cycle-cover conjecture. The graph $G$ contains five even subgraphs such that every edge of $G$ belongs to exactly two of them. This is one of the classical cycle-cover conjectures related to Petersen coloring and remains open.

   source: Vahan Mkrtchyan, “Non-conflicting no-where zero Z_2Z_2 flows in cubic graphs”, arXiv:2410.04389 (2024).