Families of Dot-Product Snarks on Orientable Surfaces of Low Genus |
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Authors: | sarah-marie Belcastro Jackie Kaminski |
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Affiliation: | (1) Department of Mathematics and Statistics, Smith College, Northampton, MA 01060, USA;(2) Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA |
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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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Keywords: | Graph genus Graph embedding Snarks Grünbaum’ s Conjecture Berge-Fulkerson Conjecture Dot product |
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