... transitions. Jour. Math. Physics,4:287–293, 1963.[7] P.W. Kasteleyn. Graph theory and crystal physics. In Graph theory and theo-retical physics, New York, 1967. Academic Press.[8] W. McCuaig, ... Bij,i =1, , g and j =1,2, wherethe electronic journal of combinatorics 6 (1999), #R6 11Moreover lD1(R2,R3,e)=lD(R2,R3,e) = 1, since r(D)d(e)= 1 and D is arelevant orientation. ... =1,2, Diis an orientation of a proper g-graph and Iiis an alternatingcycle of Di. Moreover |BR(Ii)| < |BR(C)|.Let us assume without loss of generality that h2is directed so...