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

Abstract

A graph is strongly $\Z_{\ell}$-connected if for each boundary function $β: V(G)\mapsto \Z_{\ell}$ with $β(v) \equiv d(v) \pmod{2}$ for every vertex $v$ and $\sum_{v \in V(G)} β(v) \equiv 0 \pmod{2\ell}$, there exists an orientation $D$ of $G$ such that $d_D^+(v) - d_D^-(v) \equiv β(v) \pmod{2\ell}$ 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.

Figures (2)

• Figure 1: The graphs $aK_2$, $T_{a,b,c}$.
• Figure 2: Two graphs $W_1$, and $W_2$.

Theorems & Definitions (32)

• Definition 1.1
• Proposition 1.2
• Theorem 1.3: LSWW24
• Lemma 1.4: LSWW24
• Theorem 1.5: Hakimi H65
• Definition 1.6
• Theorem 1.7: H24
• Theorem 2.1
• proof
• Claim 2.1