A Note on Zero-Sum 5-Flows in Regular Graphs

Abstract

Let $G$ be a graph. A zero-sum flow of $G$ is an assignment of non-zero real numbers to the edges of $G$ such that the sum of the values of all edges incident with each vertex is zero. Let $k$ be a natural number. A zero-sum $k$-flow is a flow with values from the set $\{\pm 1,\ldots ,\pm(k-1)\}$. It has been conjectured that every $r$-regular graph, $r\geq 3$, admits a zero-sum $5$-flow. In this paper we provide an affirmative answer to this conjecture, except for  $r=5$.