A Linear Bound on the Rich Flow Number for Graphs with a Given Maximum Degree

TL;DR

This paper establishes a linear upper bound on the rich flow number for graphs with a given maximum degree, advancing understanding of flow properties in graph theory.

Contribution

It introduces a linear bound on the rich flow number for rich flow admissible graphs based on maximum degree, which was previously unknown.

Findings

Rich flow admissible graphs with maximum degree Δ admit a rich (264Δ - 445)-flow.

The bound is linear in the maximum degree Δ.

This result improves previous bounds on rich flow numbers.

Abstract

A rich $k$-flow is a nowhere-zero $k$-flow $\phi$ such that, for every pair of adjacent edges $e$ and $f$, $|\phi(e)| \neq |\phi(f)|$. A graph is rich flow admissible if it admits a rich $k$-flow for some integer $k$. In this paper, we prove that if $G$ is a rich flow admissible graph with maximum degree $\Delta$, then $G$ admits a rich $(264\Delta - 445)$-flow.