A note on zero-sum 5-flows in regular graphs

TL;DR

This paper proves that all regular graphs with degree at least 3, except for degree 5, admit a zero-sum 5-flow, confirming a longstanding conjecture in graph theory.

Contribution

The paper provides a proof that all r-regular graphs with r≥3, except r=5, have a zero-sum 5-flow, advancing understanding of flow properties in regular graphs.

Findings

Confirmed the conjecture for r-regular graphs with r≠5

Established existence of zero-sum 5-flows in most regular graphs

Identified the exception at r=5 for zero-sum 5-flows

Abstract

Let $G$ be a graph. A zero-sum flow in $G$ is an assignment of nonzero real number to the edges such that the sum of the values of all edges incident with each vertex is zero. Let $k$ be naturel number. A zero-sum $k$-flow is a flow with value from the set $\{\pm 1, \pm 2,..., \pm(k-1)\}$. It has been conjectured that every $r$-regular graph, $r\geq 3$, admits a zero-sum 5-flow. In this paper we give an affirmative answer to this conjecture, exept for r=5.