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Uniquely edge-3-colorable graphs and snarks. (English) Zbl 0966.05027
Let \(G\) be a 3-regular simple graph that has exactly one 1-factorization. It is proved that if \(G\) is cyclically 4-edge-connected, but not cyclically 5-edge-connected, then it contains a snark as a minor. This is an approach to the conjecture of C.-Q. Zhang [J. Graph Theory 20, No. 1, 91-99 (1995; Zbl 0854.05070)] that if \(G\) is triangle-free, then it must have the Petersen graph as a minor. A weaker conjecture due to S. Fiorini and R. J. Wilson [Research Notes in Mathematics 16 (1977; Zbl 0421.05023); Selected topics in graph theory, 103-126 (1978; Zbl 0435.05024)] claims that if \(G\) is planar, then \(G\) contains a triangle. It is proved in this paper, that every counterexample to the conjecture is cyclically 5-edge-connected and that in a minimal counterexample every 5-edge-cut is trivial.

MSC:
05C15 Coloring of graphs and hypergraphs
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C75 Structural characterization of families of graphs
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