Nowhere-zero $3$-flow and $\mathbb{Z}_3$-connectedness in Graphs with Four Edge-disjoint Spanning Trees

TL;DR

This paper proves that graphs with four edge-disjoint spanning trees are $ ext{Z}_3$-connected, advancing understanding of graph orientations and connectivity related to Jaeger et al.'s conjecture.

Contribution

It establishes that any graph with four edge-disjoint spanning trees is $ ext{Z}_3$-connected, providing a new class of graphs satisfying this property.

Findings

Graphs with four edge-disjoint spanning trees are $ ext{Z}_3$-connected.

Every 5-edge-connected essentially 23-edge-connected graph is $ ext{Z}_3$-extendable at degree five.

The result supports Jaeger et al.'s conjecture for a broader class of graphs.

Abstract

Given a zero-sum function $\beta : V(G) \rightarrow \mathbb{Z}_3$ with $\sum_{v\in V(G)}\beta(v)=0$, an orientation $D$ of $G$ with $d^+_D(v)-d^-_D(v)= \beta(v)$ in $\mathbb{Z}_3$ for every vertex $v\in V(G)$ is called a $\beta$-orientation. A graph $G$ is $\mathbb{Z}_3$-connected if $G$ admits a $\beta$- orientation for every zero-sum function $\beta$. Jaeger et al. conjectured that every $5$-edge-connected graph is $\mathbb{Z}_3$-connected. A graph is $\langle\mathbb{Z}_3\rangle$-extendable at vertex $v$ if any pre-orientation at $v$ can be extended to a $\beta$-orientation of $G$ for any zero-sum function $\beta$. We observe that if every $5$-edge-connected essentially $6$-edge-connected graph is $\langle\mathbb{Z}_3\rangle$-extendable at any degree five vertex, then the above mentioned conjecture by Jaeger et al. holds as well. Furthermore, applying the partial flow extension method of Thomassen and of Lov\'{a}sz et al., we prove that every graph with at least 4 edge-disjoint spanning trees is $\mathbb{Z}_3$-connected. Consequently, every $5$-edge-connected essentially $23$-edge-connected graph is $\langle\mathbb{Z}_3\rangle$-extendable at degree five vertex.