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Generalized Connectivity of the Mycielskian Graph under g -Extra Restriction

Author

Listed:
  • Jinyu Zou

    (School of Mathematics and Physics, Qinghai University, Xining 810016, China)

  • He Li

    (School of Information Engineering, Communication University of Shanxi, Jinzhong 030619, China)

  • Shumin Zhang

    (School of Mathematics and Statistics, Qinghai Normal University, Xining 810001, China
    Academy of Plateau Science and Sustainability, People’s Government of Qinghai Province and Beijing Normal University, Beijing 100875, China)

  • Chengfu Ye

    (School of Mathematics and Statistics, Qinghai Normal University, Xining 810001, China
    Academy of Plateau Science and Sustainability, People’s Government of Qinghai Province and Beijing Normal University, Beijing 100875, China)

Abstract

The g -extra connectivity is a very important index to evaluate the fault tolerance, reliability of interconnection networks. Let g be a non-negative integer, G be a connected graph with vertex set V and edge set E , a subset S ⊆ V is called a g -extra cut of G if the graph induced by the set G − S is disconnected and each component of G − S has at least g + 1 vertices. The g-extra connectivity of G , denoted as κ g ( G ) , is the cardinality of the minimum g -extra cut of G . Mycielski introduced a graph transformation to discover chromatic numbers of triangle-free graphs that can be arbitrarily large. This transformation converts a graph G into a new compound graph called μ ( G ) , also known as the Mycielskian graph of G . In this paper, we study the relationship on g -extra connectivity between the Mycielskian graph μ ( G ) and the graph G . In addition, we show that κ 3 ( μ ( G ) ) = 2 κ 1 ( G ) + 1 for κ 1 ( G ) ≤ m i n { 4 , ⌊ n 2 ⌋ } , and prove the bounds of κ 2 g + 1 ( μ ( G ) ) for g ≥ 2 .

Suggested Citation

  • Jinyu Zou & He Li & Shumin Zhang & Chengfu Ye, 2023. "Generalized Connectivity of the Mycielskian Graph under g -Extra Restriction," Mathematics, MDPI, vol. 11(19), pages 1-11, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4043-:d:1246407
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