Linear Operators That Preserve the Genus of a Graph
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322-3900, USA
School of Computational Sciences, Korean Institute for Advanced Study, Seoul 02455, Korea
Department of Mathematics, Jeju National University, Jeju 63243, Korea
Author to whom correspondence should be addressed.
Received: 6 March 2019 / Revised: 21 March 2019 / Accepted: 25 March 2019 / Published: 28 March 2019
PDF [223 KB, uploaded 28 March 2019]
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
. 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
to graphs of genus
. 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.
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MDPI and ACS Style
Beasley, L.B.; Kim, J.H.; Song, S.-Z. Linear Operators That Preserve the Genus of a Graph. Mathematics 2019, 7, 312.
Beasley LB, Kim JH, Song S-Z. Linear Operators That Preserve the Genus of a Graph. Mathematics. 2019; 7(4):312.
Beasley, LeRoy B.; Kim, Jeong H.; Song, Seok-Zun. 2019. "Linear Operators That Preserve the Genus of a Graph." Mathematics 7, no. 4: 312.
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