Hamilton weights and Petersen minors

A (1, 2)-eulerian weight w of a cubic graph is called a Hamilton weight if every faithful circuit cover of the graph with respect to w is a set of two Hamilton circuits. Let G be a 3-connected cubic graph containing no Petersen minor. It is proved in this paper that G admits a Hamilton weight if and only if G can be obtained from K4 by a series of4$Y-operations. As a byproduct of the proof of the main theorem, we also prove that if G is a permutation graph and w is a (1,2)-eulerian weight of G such that (G,w) is a critical contra pair, then the Petersen minor appears ``almost everywhere'' in the graph G. ß 2001 John Wiley & Sons, Inc. J Graph Theory 38: 197±219, 2001