Zero-Sum Flow Numbers of Regular Graphs

摘要

As an analogous concept of a nowhere-zero flow for directed graphs, we consider zero-sum flows for undirected graphs in this article. For an undirected graph G, a zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum k-flow if the values of edges are less than k. We define the zero-sum flow number of G as the least integer k for which G admitting a zero-sum k-flow. In this paper, among others we calculate the zero-sum flow numbers for regular graphs and also the zero-sum flow numbers for Cartesian products of regular graphs with paths.