The (5, 2)-cycle-cover conjecture
Let 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 contains five even subgraphs such that every edge of belongs to exactly two of them.
This is a cycle-cover conjecture extending the perfect-matching formulation beyond cubic graphs. The source attributes it to Seymour and Szekeres, and its resolution is not supplied in the source.
Equivalent formulations 2Other wordings
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
-cycle-cover conjecture
Let be a bridgeless graph, not necessarily cubic. An even subgraph is a subgraph in which every vertex of has even degree. -cycle-cover conjecture. The graph 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).
The -cycle-cover conjecture
Let be a bridgeless graph, not necessarily cubic. An even subgraph is a subgraph in which every vertex has even degree. The -cycle-cover conjecture. The graph contains five even subgraphs such that every edge of 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).
References
Primary source
Anush Hakobyan and Vahan Mkrtchyan, “On Sylvester Colorings of Cubic Graphs”, arXiv:1511.02475 (2018).
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