Integer Functions on the Cycle Space and Edges of a Graph

Abstract

A directed graph has a natural $${\mathbb {Z}}$$-module homomorphism from the underlying graph’s cycle space to $${\mathbb {Z}}$$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 $${\mathbb {Z}}$$-module homomorphism of a graph’s cycle space to $${\mathbb {Z}}$$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.