Zero-sum 6-flows in 5-regular graphs

TL;DR

This paper proves that every 5-regular graph admits a zero-sum 6-flow, meaning edges can be assigned values from b1a01 to b1a05 such that the sum at each vertex is zero.

Contribution

The paper establishes that all 5-regular graphs have a zero-sum 6-flow, advancing understanding of flow properties in regular graphs.

Findings

Every 5-regular graph admits a zero-sum 6-flow.

Zero-sum flows can be assigned with values from b1a01 to b1a05.

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