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

Abstract

A graph is strongly ${\mathbb{Z}}_{\ell }$-connected if for each boundary function ${\hat{I}}^2:V(G)\mapsto {\mathbb{Z}}_{\ell }$ with ${\hat{I}}^2(v)\equiv d(v)(\mathrm{mod}2)$ for every vertex $v$ and ${\sum }_{v\in V(G)}{\hat{I}}^2(v)\equiv 0(\mathrm{mod}2\ell )$, there exists an orientation $D$ of $G$ such that $d_D^{+}(v)-d_D^{-}(v)\equiv {\hat{I}}^2(v)(\mathrm{mod}2\ell )$ 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 $\ell \ge 2$, which will be used in our further study in this topic.