Linear Operators That Preserve Two Genera of a Graph

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 li...

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Bibliographic Details
Published in:Mathematics (Basel) 2020-05, Vol.8 (5), p.676
Main Authors: Beasley, LeRoy B., Kang, Kyung-Tae, Song, Seok-Zun
Format: Article
Language:English
Subjects:
genus of a graph
linear operator
vertex permutation
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math8050676