On density of $Z_3$-flow-critical graphs

TL;DR

This paper establishes a density bound for $Z_3$-flow-critical graphs on fixed surfaces, extending known planar results to more general surface embeddings, and contributing to graph flow theory.

Contribution

It generalizes the density bounds of $Z_3$-flow-critical graphs from planar graphs to graphs on arbitrary surfaces.

Findings

Derived a density bound for $Z_3$-flow-critical graphs on fixed surfaces.

Extended planar graph flow results to surface-embedded graphs.

Provided a theoretical framework for analyzing flow-critical graphs on surfaces.

Abstract

For an abelian group $\Gamma$, a graph $G$ is said to be $\Gamma$-flow-critical if $G$ does not admit a nowhere-zero $\Gamma$-flow, but for each edge $e\in E(G)$, the contraction $G/e$ has a nowhere-zero $\Gamma$-flow. A bound on the density of $Z_3$-flow-critical graphs drawn on a fixed surface is obtained, generalizing the planar case of the bound on the density of 4-critical graphs by Kostochka and Yancey.