0-Sum and 1-Sum Flows in Regular Graphs

Abstract

Let $G$ be a graph. Assume that $l$ and $k$ are two natural numbers. An $l$-sum flow on a graph $G$ is an assignment of non-zero real numbers to the edges of $G$ such that for every vertex $v$ of $G$ the sum of values of all edges incidence with $v$ equals $l$. An $l$-sum $k$-flow is an $l$-sum flow with values from the set $\{\pm 1,\ldots ,\pm(k-1)\}$. Recently, it was proved that for every $r, r\geq 3$, $r\neq 5$, every $r$-regular graph admits a $0$-sum $5$-flow. In this paper we settle a conjecture by showing that every $5$-regular graph admits a $0$-sum $5$-flow. Moreover, we prove that every $r$-regular graph of even order admits a $1$-sum $5$-flow.