ABSTRACT Cun-Quan Zhang DEPARTMENT OF MATHEMA TICS WEST VIRGINIA UNIVERSITY MORGANTOWN, WEST VIRGINIA A (1,2)-eulerian weight w of a graph is hamiltonian if every faithful cover of w is a set of two Hamilton circuits. Let G be a 3-connected cubic graph containing no subdivision of the Petersen graph. We prove that if G admits a hamiltonian weight then G is uniquely 3-edge-colorable. © 1995 John Wiley & Sons, Inc. 1. INTRODUCTION Most standard graph-theoretic tenns that are used in this paper can be found for instance in [3]. All graphs we considered in this paper may have multiple edges but no loops. A (1, 2)-eulerian weight w of a 2-connected graph G is a weight w: E(G) 1-')0 {l, 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. The topic of faithful coverings of eulerian weights has attracted many mathematicians (see survey papers [6-9,19], etc.). Some well-known conjectures on this topic (such as the circuit double cover conjecture due to Szekeres [12] and, Seymour [11]) still remain open. A cubic graph is uniquely 3-edge-colorable if G has precisely one I-factorization. The topic of uniquely 3-edge-colorable cubic graphs is also a very interesting topic in graph theory (see [13,14,17,4, and 10], etc.). Some well-known conjectures (such as every planar, uniquely 3-edge­colorable, cubic graph has a triangle due to Fiorini and Wilson [4]) still remain open. In this paper, we will study the relations between the faithful coverings and the uniquely 3-edge-colorability of cubic graphs.
['Cun-Quan Zhang']
Hamiltonian weights and unique 3-edge-colorings of cubic graphs