The (5,2)-even-subgraph-cover conjecture
Let be a bridgeless graph, not necessarily cubic. An even subgraph is a subgraph in which every vertex has even degree.
The -even-subgraph-cover conjecture. There exist five even subgraphs of such that every edge of belongs to exactly two of them.
This conjecture is related to cycle-cover problems and is used in the paper to derive consequences for cubic-graph colorings. Its status is not resolved in the supplied text.
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Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
The -even-subgraph-cover conjecture
Let be a bridgeless graph, not necessarily cubic. An even subgraph is a subgraph in which every vertex has even degree. -even-subgraph-cover conjecture. The graph contains five even subgraphs such that every edge of belongs to exactly two of them.
This conjecture is presented as a classical consequence of the Petersen coloring conjecture. The source gives no evidence that it has been resolved.
source: Luca Ferrarini and Vahan Mkrtchyan, “Some new results on Sylvester colorings of cubic graphs”, arXiv:2607.06396 (2026).
References
Primary source
Anush Hakobyan and Vahan Mkrtchyan, “S_12 and P_12-colorings of cubic graphs”, arXiv:1807.08138 (2018).
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