Hamilton weight and Petersen minor

A (1, 2)-eulerian weight w of a cubic graph is hamiltonian 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 of 4 ↔ 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.