
S. Akbari

M. Kano

S. Zare
Keywords:
$0$Sum Flow, Regular Graph, $1$Sum Flow, Factor
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 nonzero 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 $\{\pm 1,\ldots ,\pm(k1)\}$. 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.