Zero-sum 6-flows in 5-regular graphs

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 $\{\pm1, \ldots, \pm(k - 1)\}$. In this paper, we prove that every 5-regular graph admits a zero-sum 6-flow.