Families of dot-product snarks on orientable surfaces of low genus

Sarah Marie Belcastro, Jaclyn Nicole Kaminski

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

We introduce a generalized dot product and provide some embedding conditions under which the genus of a graph does not rise under a dot product with the Petersen graph. Using these conditions, we disprove a conjecture of Tinsley and Watkins on the genus of dot products of the Petersen graph and show that both Grünbaum's Conjecture and the Berge-Fulkerson Conjecture hold for certain infinite families of snarks. Additionally, we determine the orientable genus of four known snarks and two known snark families, construct a new example of an infinite family of snarks on the torus, and construct ten new examples of infinite families of snarks on the 2-holed torus; these last constructions allow us to show that there are genus-2 snarks of every even order n ≥ 18.

Original languageEnglish (US)
Pages (from-to)229-240
Number of pages12
JournalGraphs and Combinatorics
Volume23
Issue number3
DOIs
StatePublished - Jun 1 2007

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Snark
Scalar, inner or dot product
Genus
Petersen Graph
Torus
Disprove
Family
Graph in graph theory

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

Cite this

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Families of dot-product snarks on orientable surfaces of low genus. / Belcastro, Sarah Marie; Kaminski, Jaclyn Nicole.

In: Graphs and Combinatorics, Vol. 23, No. 3, 01.06.2007, p. 229-240.

Research output: Contribution to journalArticle

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