Cyclically $5$-edge-connected snarks with resistance $2$ and flow resistance $n$

TL;DR

This paper constructs a family of highly connected snarks with resistance and flow resistance properties, demonstrating that their ratio can be made arbitrarily large, addressing a recent open question.

Contribution

It provides the first explicit construction of cyclically 5-edge-connected snarks with unbounded flow resistance to resistance ratio.

Findings

Constructed snarks with resistance 2 and arbitrarily large flow resistance.

Showed the ratio of flow resistance to resistance can be made arbitrarily large.

Answered a recent open question in graph theory.

Abstract

Snarks are $2$-connected cubic graphs that do not admit a proper $3$-edge-coloring. For a cubic graph $G$, its resistance $r(G)$ is the minimum number of edges whose removal results in a $3$-edge-colorable graph, while its flow resistance $r_f(G)$ is the minimum number of edges whose removal results in a graph admitting a nowhere-zero $\mathbb{Z}_2 \times \mathbb{Z}_2$-flow. In this paper, we provide an affirmative answer to a question recently posed by Allie, M\'a\v{c}ajov\'a, and \v{S}koviera by constructing a family of cyclically $5$-edge-connected snarks for which the ratio $r_f(G)/r(G)$ is arbitrarily large.