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.
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Belcastro, sm., Kaminski, J. Families of Dot-Product Snarks on Orientable Surfaces of Low Genus. Graphs and Combinatorics 23, 229–240 (2007). https://doi.org/10.1007/s00373-007-0729-9
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DOI: https://doi.org/10.1007/s00373-007-0729-9