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 K 4 by a series of Δ↔ 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 . © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 197–219, 2001