Cycle covers (I) – Minimal contra pairs and Hamilton weights

Let G be a bridgeless cubic graph associated with an eulerian weight w:E(G)↦{1,2}. A faithful circuit cover of the pair (G,w) is a family of circuits in G which covers each edge e of G precisely w(e) times. A circuit C of G is removable if the graph obtained from G by deleting all weight 1 edges contained in C remains bridgeless. A pair (G,w) is called a contra pair if it has no faithful circuit cover, and a contra pair (G,w) is minimal if it has no removable circuit, but for each weight 2 edge e, the graph G−e has a faithful circuit cover with respect to the weight w. It is proved by Alspach et al. (1994) [2], that if (G,w) is a minimal contra pair, then the graph G must contain a Petersen minor. It is further conjectured by Fleischner and Jackson (1988) [5] that this graph G must be the Petersen graph itself (not just as a minor). In this paper, we prove that this conjecture is true if every Hamilton weight graph is constructed from K4 via a series of (Y→△)-operations.