Integer Functions on the Cycle Space and Edges of a Graph

A directed graph has a natural mathbb Z{mathbb {Z}} -module homomorphism from the underlying graph’s cycle space to mathbb Z{mathbb {Z}} where the image of an oriented cycle is the number of forward edges minus the number of backward edges. Such a homomorphism preserves the parity of the length of a cycle and the image of a cycle is bounded by the length of that cycle. Pretzel and Youngs (SIAM J. Discrete Math. 3(4):544–553, 1990) showed that any mathbb Z{mathbb {Z}} -module homomorphism of a graph’s cycle space to mathbb Z{mathbb {Z}} that satisfies these two properties for all cycles must be such a map induced from an edge direction on the graph. In this paper we will prove a generalization of this theorem and an analogue as well.