On the assignment of loop sets to graphs

It is shown that it is always possible to assign m+1 loop sets to a graph, planar or nonplanar, of nullity m so that the number of loop sets traversing in one direction through any edge equals the number of loop sets traversing in the opposite direction of the edge. Furthermore, any m of those m+1 loop sets are linearly independent. The authors present a constructive proof of this result. The proof itself serves as an algorithm for the enumeration of the desired loop sets