Characterization of strongly $\mathbb{Z}_\ell$-connected graphs of small order

TL;DR

This paper provides a complete, manual proof characterizing all 4-vertex graphs that are strongly _\u03bb-connected for any integer 2, aiding the study of circular flows in graphs.

Contribution

It offers a self-contained proof characterizing 4-vertex strongly _-connected graphs for all   2, which is novel and useful for circular flow research.

Findings

Characterization of 4-vertex strongly _-connected graphs for all   2

Manual proof method for the characterization

Foundation for further studies in circular flows

Abstract

A graph is strongly $\Z_{\ell}$-connected if for each boundary function $\beta: V(G)\mapsto \Z_{\ell}$ with $\beta(v) \equiv d(v) \pmod{2}$ for every vertex $v$ and $\sum_{v \in V(G)} \beta(v) \equiv 0 \pmod{2\ell}$, there exists an orientation $D$ of $G$ such that $d_D^+(v) - d_D^-(v) \equiv \beta(v) \pmod{2\ell}$ for each $v \in V(G)$. This is a useful notion for studying circular flows of graphs. This note presents a fully self-contained, manual proof of a characterization of $4$-vertex strongly $\mathbb{Z}_\ell$-connected graphs for any integer $\ell\geq 2$, which will be used in our further study in this topic.