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

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, Máčajová, and Škoviera by constructing a family of cyclically $5$-edge-connected snarks for which the ratio $r_f(G)/r(G)$ is arbitrarily large.

• The paper presents an explicit construction of cyclically 5-edge-connected snarks with fixed resistance 2 and arbitrarily high flow resistance n.
• It uses inductive methods involving modified Petersen graph substructures and recursive semi-graph compositions to enforce strict colorability constraints.
• The findings challenge previous conjectures by demonstrating that the ratio of flow resistance to resistance can be made arbitrarily large.

Cyclically 5-Edge-Connected Snarks Exhibiting Arbitrary Ratios of Flow Resistance to Resistance

Introduction

The paper addresses a fundamental question in the study of cubic graphs, specifically snarks—2-connected, 3-regular graphs not admitting proper 3-edge-coloring. The interplay between two key parameters, resistance ($r(G)$) and flow resistance ($rf(G)$), measures the minimal edge removal necessary to render a snark 3-edge-colorable or to permit a nowhere-zero $Z_2 \times Z_2$ flow, respectively. While these parameters are closely related, prior work established their divergence in certain snarks. The paper resolves a question posed by Allie, Máčajová, and Škoviera regarding whether the ratio $rf(G)/r(G)$ can be arbitrarily large in cyclically 5-edge-connected snarks. The authors construct an infinite family of such graphs, each denoted $H_n$, with $r(H_n) = 2$, $rf(H_n) = n$, and cyclic 5-edge-connectivity.

Preliminaries and Key Concepts

The paper relies on advanced concepts from structural graph theory:

• Snarks: Cubic, 2-connected graphs not 3-edge-colorable.
• Resistance ($r(G)$): Minimum number of edges removed to obtain a 3-edge-colorable subgraph.
• Flow resistance ($rf(G)$): Minimum edge removal enabling a nowhere-zero $Z_2 \times Z_2$ flow.
• Cyclically $rf(G)$0-edge-connected: No cyclic edge-cut in $rf(G)$1 of size less than $rf(G)$2.

The authors utilize semi-graphs (graphs allowing semi-edges) to facilitate recursive constructions and colorings. The parity lemma and properties of nowhere-zero flows are central in their arguments, with interaction between edge-colorings and abelian-group flows (notably $rf(G)$3) providing bridges in the analysis.

Constructive Methodology

The construction is inductive, leveraging modified substructures of the Petersen graph (the classical snark prototype):

1. Base Semi-graphs ($rf(G)$4, $rf(G)$5, $rf(G)$6): The semi-graphs $rf(G)$7 and $rf(G)$8 are crafted from the Petersen graph by vertex/edge removals, maintaining cubicity and controlling colorability. $rf(G)$9 is assembled by integrating $Z_2 \times Z_2$0 and $Z_2 \times Z_2$1, with careful merging of semi-edges such that strong color constraints are enforced.
2. Recursive Family ($Z_2 \times Z_2$2): A sequence $Z_2 \times Z_2$3 is defined, starting from $Z_2 \times Z_2$4 and building each $Z_2 \times Z_2$5 via merging copies of $Z_2 \times Z_2$6 and $Z_2 \times Z_2$7, preserving properties essential to resistance and flow resistance. The coloring and flow arguments ensure that resistance remains strictly 1 throughout recursion, while flow resistance increments at each stage.
3. Final Snark ($Z_2 \times Z_2$8): The snark $Z_2 \times Z_2$9 is obtained by adding vertices and edges to $rf(G)/r(G)$0, achieving cubicity, cyclic 5-edge-connectivity, and solidifying the desired resistance and flow resistance properties.

Analytical Results

The paper rigorously proves several structural and coloring properties:

• Resistance and Flow Resistance: In $rf(G)/r(G)$1 and subsequently $rf(G)/r(G)$2, proper colorings are blocked except at one or two vertices, yielding $rf(G)/r(G)$3. Inductive arguments, aided by the parity lemma and analysis of Kempe chains, show that the minimal number of edge removals for a nowhere-zero $rf(G)/r(G)$4 flow is precisely $rf(G)/r(G)$5—i.e., $rf(G)/r(G)$6.
• Cyclic 5-Edge-Connectivity: Through recursive application of graph operations and leveraging connectivity lemmas, it is shown that $rf(G)/r(G)$7 maintains cyclic 5-edge-connectivity for all $rf(G)/r(G)$8.
• Ratio of Flow Resistance to Resistance: The family exhibits $rf(G)/r(G)$9, which becomes arbitrarily large as $H_n$0 increases—addressing and affirmatively settling the outstanding conjecture for cyclically 5-edge-connected snarks.

Numerical Values and Contradictory Claims

The principal numerical result is as follows:

• Order of $H_n$1: Each $H_n$2 has $H_n$3 vertices.
• Resistance: $H_n$4 (minimum possible for a snark).
• Flow resistance: $H_n$5, unbounded as $H_n$6 grows.

A contradictory claim relative to prior conjectures is established: contrary to Fiol, Mazzuoccolo, and Steffen's conjecture that flow resistance should not exceed resistance in bridgeless cubic graphs, these constructions permit the ratio to be arbitrarily large.

Implications and Future Directions

The construction decisively demonstrates that resistance and flow resistance are independent structural parameters in snarks, amplifying the complexity underlying colorability and flow properties in cubic graphs. This result:

• Impacts Conjecture Reductions: As smallest counterexamples to major conjectures must be snarks, understanding their resistance spectrum is critical.
• Challenges Bounds: Practitioners must now account for unbounded ratios of these parameters in algorithmic and combinatorial analyses.
• Suggests Structural Diversity: The possibility of highly connected snarks with low resistance and high flow resistance points to rich morphologies within cubic graph families, with potential ramifications for graph flow theory and computational approaches.

Further exploration is warranted for classification, enumeration, and automorphism profiles of such snarks, as well as for potential refinement of connectivity and colorability conjectures.

Conclusion

This paper delivers a meticulous construction of an infinite family of cyclically 5-edge-connected snarks with resistance 2 and arbitrary flow resistance, conclusively resolving an open question in cubic graph theory. The interplay between resistance, flow resistance, and high cyclic connectivity is examined with technical rigor. These results extend the boundaries of known snark properties, refining theoretical understanding and imposing new constraints on graph colorability and flow conjectures.