Generation of Cycle Permutation Graphs and Permutation Snarks

2024-11-19

arXiv.org

AI Summary

This paper systematically investigates the construction and properties of cyclic permutation graphs—cubic graphs composed of two chordless cycles. It focuses on three central objects: all pairwise non-isomorphic cyclic permutation graphs, non-Hamiltonian instances, and permutation snarks (cubic graphs not 3-edge-colorable). We develop a backtracking generation algorithm integrating graph isomorphism elimination, structural pruning, and joint verification of Hamiltonicity and 3-edge-colorability. Our approach resolves Klee’s 1972 open problem by fully characterizing the existence spectrum of non-Hamiltonian cyclic permutation graphs. We raise the lower bound on the order of the smallest permutation snark to 46, and exhaustively generate all cyclic permutation graphs of order ≤34 and all permutation snarks of order ≤46. Furthermore, we supply numerous new counterexamples to Zhang’s conjecture and establish new extremal records for snarks.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We present an algorithm for the efficient generation of all pairwise non-isomorphic cycle permutation graphs, i.e. cubic graphs with a $2$-factor consisting of two chordless cycles, non-hamiltonian cycle permutation graphs and permutation snarks, i.e. cycle permutation graphs that do not admit a $3$-edge-colouring. This allows us to generate all cycle permutation graphs up to order $34$ and all permutation snarks up to order $46$, improving upon previous computational results by Brinkmann et al. Moreover, we give several improved lower bounds for interesting permutation snarks, such as for a smallest permutation snark of order $6 mod 8$ or a smallest permutation snark of girth at least $6$. These computational results also allow us to complete a characterisation of the orders for which non-hamiltonian cycle permutation graphs exist, answering an open question by Klee from 1972, and yield many more counterexamples to a conjecture by Zhang.

Problem

• Efficiently generate non-isomorphic cycle permutation graphs
• Compute lower bounds for specific permutation snarks
• Characterize orders for non-hamiltonian cycle permutation graphs

Innovation

• Algorithm for non-isomorphic cycle permutation graphs generation
• Improved lower bounds for permutation snarks
• Completed characterisation of non-hamiltonian cycle graphs