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Graphs and Combinatorics

, Volume 16, Issue 3, pp 257–267 | Cite as

Uniquely Edge-3-Colorable Graphs and Snarks

  • 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|).

Keywords

Petersen Graph Minimal Counterexample 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Tokyo 2000

Authors and Affiliations

  • John L. Goldwasser
    • 1
  • Cun-Quan Zhang
    • 1
  1. 1.Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA.US

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