A Linear Bound on the Rich Flow Number for Graphs with a Given Maximum Degree
Robert Lukoťka
TL;DR
The paper addresses the problem of bounding the rich flow number for graphs with a given maximum degree. It develops a constructive approach based on $(\mathbb{Z}_k \times \mathbb{Z}_2)$-flows (with $k = 8\Delta - 13$), organized around circuit chains and careful control of confluent/contrafluent adjacent edge pairs, and uses a 2-edge-cut decomposition along with Seymour's flow techniques to assemble a global rich flow. The main result is a linear upper bound: every rich flow admissible graph with maximum degree $\Delta$ admits a rich $(264\Delta - 445)$-flow, implemented by combining three auxiliary flows and a modular linear combination. This advances the understanding of how flow constraints interact with degree parameters and suggests directions for tightening constants and exploring related conjectures for specific connectivity or relaxations of the nowhere-zero condition.
Abstract
A rich $k$-flow is a nowhere-zero $k$-flow $φ$ such that, for every pair of adjacent edges $e$ and $f$, $|φ(e)| \neq |φ(f)|$. 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 $Δ$, then $G$ admits a rich $(264Δ- 445)$-flow.