TL;DR
This paper computes the design spectra for all non-trivial snarks up to 22 vertices, including several well-known graphs, and provides partial results for larger snarks, advancing understanding of their combinatorial properties.
Contribution
It offers the first comprehensive spectrum analysis for all non-trivial snarks up to 22 vertices and extends partial results to larger, more complex snarks.
Findings
Complete spectra for all non-trivial snarks up to 22 vertices.
Partial spectrum results for larger snarks like the Celmins-Swart and Blanusa snarks.
Enhanced understanding of the combinatorial design properties of snarks.
Abstract
The main aim of this paper is to solve the design spectrum problem for Tietze's graph, the two 18-vertex Blanusa snarks, the six snarks on 20 vertices (including the flower snark J5), the twenty snarks on 22 vertices (including the two Loupekine snarks) and Goldberg's snark 3. Together with the Petersen graph (for which the spectrum has already been computed) this list includes all non-trivial snarks of up to 22 vertices. We also give partial results for a selection of larger graphs: the two Celmins-Swart snarks, the 26- and 34-vertex Blanusa snarks, the flower snark J7, the double star snark, Zamfirescu's graph, Goldberg's snark 5, the Szekeres snark and the Watkins snark.
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Taxonomy
Topicsgraph theory and CDMA systems · Finite Group Theory Research · Graph Labeling and Dimension Problems