Graphs and Combinatorics

, Volume 23, Issue 3, pp 229–240

Families of Dot-Product Snarks on Orientable Surfaces of Low Genus

Article

DOI: 10.1007/s00373-007-0729-9

Cite this article as:
Belcastro, . & Kaminski, J. Graphs and Combinatorics (2007) 23: 229. doi:10.1007/s00373-007-0729-9

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.

Keywords

Graph genus Graph embedding Snarks Grünbaum’s Conjecture Berge-Fulkerson Conjecture Dot product 

Copyright information

© Springer-Verlag Tokyo 2007

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSmith CollegeNorthamptonUSA
  2. 2.Department of Mathematical SciencesBinghamton UniversityBinghamtonUSA