Integer Functions on the Cycle Space and Edges of a Graph

A directed graph has a natural Z-module homomorphism from the underlying graph’s cycle space to 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 [1] showed that any Z-module homomorphism of a graph’s cycle space to 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.

• Type: Journal
• Year: 2010
• Where: GC
• Authors: Daniel C. Slilaty
• Keywords: Applied Computing, GC 2010, Graph’s Cycle Space, Natural Z-module Homomorphism, Z-module Homomorphism