Abstract:A cubic graph G is uniquely edge-3-colorable if G has precisely one 1-factorization. It is proved in this paper, if a uniquely edge-3-colorable, cubic graph G is cyclically 4-edgeconnected, but not cyclically 5-edge-connected, then G must contain a snark as a minor. This is an approach to a conjecture that every triangle free uniquely edge-3-colorable cubic graph must have the Petersen graph as a minor. Fiorini and Wilson (1976) conjectured that every uniquely edge-3-colorable planar cubic graph must have a t… Show more
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