| Let w: E(G) →{1,2}. A faithful circuit cover C of G with respect to w is a family of circuits such that every edge e is contained in precisely w(e) members of C. A weight w is a Hamilton weight of G if every faithful circuit cover of G with respect to w is a set of two Hamilton circuits. The study of Hamilton weight is motivated by the conjecture of circuit double cover. It was conjectured that every 3-connected cubic graph admitting a Hamilton weight is obtained from K by a series of Y－-operations. The conjecture is proved for Petersen-minor-free graphs. And this conjecture also implies a conjecture by Fleischner and Jackson for compatible circuit decomposition. |