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.
@article{10_37236_4986,
author = {S. Akbari and M. Kano and S. Zare},
title = {0-sum and 1-sum flows in regular graphs},
journal = {The electronic journal of combinatorics},
year = {2016},
volume = {23},
number = {2},
doi = {10.37236/4986},
zbl = {1338.05106},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4986/}
}
TY - JOUR
AU - S. Akbari
AU - M. Kano
AU - S. Zare
TI - 0-sum and 1-sum flows in regular graphs
JO - The electronic journal of combinatorics
PY - 2016
VL - 23
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/4986/
DO - 10.37236/4986
ID - 10_37236_4986
ER -
%0 Journal Article
%A S. Akbari
%A M. Kano
%A S. Zare
%T 0-sum and 1-sum flows in regular graphs
%J The electronic journal of combinatorics
%D 2016
%V 23
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/4986/
%R 10.37236/4986
%F 10_37236_4986
S. Akbari; M. Kano; S. Zare. 0-sum and 1-sum flows in regular graphs. The electronic journal of combinatorics, Tome 23 (2016) no. 2. doi: 10.37236/4986
