Linear Operators That Preserve the Genus of a Graph

Authors

LeRoy B. Beasley, Utah State UniversityFollow
Jeong Han Kim, Korean Institute for Advanced Study
Seok-Zun Song, Jeju National University

Document Type

Article

Journal/Book Title/Conference

Mathematics

Volume

7

Issue

4

Publisher

M D P I AG

Publication Date

3-28-2019

First Page

1

Last Page

6

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

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.

Recommended Citation

Beasley, L.B.; Kim, J.H.; Song, S.-Z. Linear Operators That Preserve the Genus of a Graph. Mathematics 2019, 7, 312.