Constant Sum Flows in Regular Graphs

Abstract

For an undirected graph G, a zero-sum flow is an assignment of non-zero integers to the edges such that the sum of the values of all edges incident with each vertex is zero. We extend this notion to a more general one in this paper, namely a constant-sum flow. The constant under a constant-sum flow is called an index of G, and I(G) is denoted as the index set of all possible indices of G. Among others we obtain that the index set of a regular graph admitting a perfect matching is the set of all integers. We also completely determine the index sets of all r-regular graphs except that of 4k-regular graphs of even order, k ≥ 1.