### 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 $r=5$.#### Keywords

Zero-sum flow, regular graph