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Hamiltonian weights and unique 3-edge-colorings of cubic graphs. (English) Zbl 0854.05070
A $$(1, 2)$$-eulerian weight $$w$$ of a 2-connected graph $$G$$ is a weight $$w: E(G)\to \{1, 2\}$$ such that the total weight of each edge-cut is even. A faithful cover of $$w$$ is a family $$C$$ of circuits such that each edge $$e$$ is contained in precisely $$w(e)$$ circuits of $$C$$. A $$(1, 2)$$-eulerian weight $$w$$ of a graph is hamiltonian if every faithful cover of $$w$$ is a set of two Hamilton circuits. A cubic graph is uniquely 3-edge-colorable if the graph has precisely one 1-factorization. The topic of faithful coverings is related to the circuit double cover conjecture and the topic of uniquely 3-edge-colorable cubic graphs is also an important subject in graph theory. Relation between the faithful coverings and the uniquely 3-edge-colorability of cubic graphs is studied in this paper and it is proved that if a 3-connected cubic graph $$G$$ containing no subdivision of the Petersen graph admits a hamiltonian weight, then $$G$$ is uniquely 3-edge-colorable.
Reviewer: H.Li (Orsay)

##### MSC:
 05C45 Eulerian and Hamiltonian graphs 05C15 Coloring of graphs and hypergraphs 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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