CIRCULAR ZERO-SUM FLOW OF REGULAR GRAPHS

A circular zero-sum r-flow for a graph G, is an assignment of real numbers from [−r + 1, −1] ∪ [1, r − 1] to the edges of G, such that for every vertex v, the sum of numbers assigned to edges incident with v is zero. The minimum possible value of r is called the circular zero-sum flow number of G. In this paper, we investigate circular zero-sum flows of regular graphs. In particular, we find the circular zero-sum flow number of even-regular graphs and provide lower and upper bounds on the circular zero-sum flow number of odd- regular graphs. Finally, we prove a necessary and sufficient condition for a graph G to have a circular zero-sum r-flow.