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Linear Operators That Preserve Two Genera of a Graph

Author

Listed:
  • 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)

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.

Suggested Citation

  • LeRoy B. Beasley & Kyung-Tae Kang & Seok-Zun Song, 2020. "Linear Operators That Preserve Two Genera of a Graph," Mathematics, MDPI, vol. 8(5), pages 1-8, April.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:5:p:676-:d:352318
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