caa a22 4500 475778103 CHVBK 20180406123636.0 cr unu---uuuuu 170329e20000901xx s 000 0 eng 10.1007/PL00007221 doi (NATIONALLICENCE)springer-10.1007/PL00007221 Uniquely Edge-3-Colorable Graphs and Snarks [Elektronische Daten] [John L. Goldwasser, Cun-Quan Zhang] 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-edge-connected, 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 triangle. It is proved in this paper that every counterexample to this conjecture is cyclically 5-edge-connected and that in a minimal counterexample to the conjecture, every cyclic 5-edge-cut is trivial (an edge-cut T of G is cyclic if no component of G\T is acyclic and a cyclic edge-cut T is trivial if one component of G\T is a circuit of length |T|). Springer-Verlag Tokyo, 2000 Goldwasser John L. Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA., US aut Zhang Cun-Quan Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA., US aut https://doi.org/10.1007/PL00007221 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/PL00007221 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Goldwasser John L. Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA., US aut NATIONALLICENCE 700 1- Zhang Cun-Quan Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA., US aut Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer