Group Connectivity: $\mathbb Z_4$ v. $\mathbb Z_2^2$

TL;DR

This paper investigates the relationship between $ ext{Z}_4$-connectivity and $ ext{Z}_2^2$-connectivity in graphs, demonstrating they are not equivalent through computationally verified examples, highlighting the complexity of group connectivity properties.

Contribution

The authors provide the first known examples showing that $ ext{Z}_2^2$-connectivity does not imply $ ext{Z}_4$-connectivity and vice versa, using computational methods.

Findings

$ ext{Z}_2^2$-connectivity does not imply $ ext{Z}_4$-connectivity

$ ext{Z}_4$-connectivity does not imply $ ext{Z}_2^2$-connectivity

Small graphs with specific properties certify these results

Abstract

We answer a question on group connectivity suggested by Jaeger et al. [Group connectivity of graphs -- A nonhomogeneous analogue of nowhere-zero flow properties, JCTB 1992]: we find that $\mathbb Z_2^2$-connectivity does not imply $\mathbb Z_4$-connectivity, neither vice versa. We use a computer to find the graphs certifying this and to verify their properties using non-trivial enumerative algorithm. While the graphs are small (the largest has 15 vertices and 21 edges), a computer-free approach remains elusive.