Linear Operators That Preserve the Genus of a Graph

LeRoy B. Beasley, Jeong Han Kim and Seok-Zun Song
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LeRoy B. Beasley: Department of Mathematics and Statistics, Utah State University, Logan, UT 84322-3900, USA
Jeong Han Kim: School of Computational Sciences, Korean Institute for Advanced Study, Seoul 02455, Korea
Seok-Zun Song: Department of Mathematics, Jeju National University, Jeju 63243, Korea

Mathematics, 2019, vol. 7, issue 4, 1-6

Abstract: A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k − 1 . A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and if it maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus k + 1 to graphs of genus k + 1 . We show that such linear operators are necessarily vertex permutations. Similar results with different restrictions on the genus k preserving operators give the same conclusion.

Keywords: linear operator; genus of a graph; vertex permutation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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