Linear Operators That Preserve Two Genera of a Graph

LeRoy B. Beasley, Kyung-Tae Kang and Seok-Zun Song
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LeRoy B. Beasley: Department of Mathematics and Statistics, Utah State University, Logan, UT 84322-3900, USA
Kyung-Tae Kang: Department of Mathematics, Jeju National University, Jeju 63243, Korea
Seok-Zun Song: Department of Mathematics, Jeju National University, Jeju 63243, Korea

Mathematics, 2020, vol. 8, issue 5, 1-8

Abstract: If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and can not be embedded on one of genus g − 1 without edge crossings, then we say that the graph has genus g . We consider a mapping on the set of graphs with m vertices into itself. The mapping is called a linear operator if it preserves a union of graphs and it also preserves the empty graph. On the set of graphs with m vertices, we consider and investigate those linear operators which map graphs of genus g to graphs of genus g and graphs of genus g + j to graphs of genus g + j for j ≤ g and m sufficiently large. We show that such linear operators are necessarily vertex permutations.

Keywords: genus of a graph; linear operator; vertex permutation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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