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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)

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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