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The Orbits of Twisted Crossed Cubes

Author

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  • Jia-Jie Liu

    (Department of Information Management, Shih Hsin University, 1 Lane 17 Sec.1, Muzha Rd., Wenshan Dist., Taipei 116, Taiwan)

Abstract

Two vertices u and v in a graph G = ( V , E ) are in the same orbit if there exists an automorphism ϕ of G such that ϕ ( u ) = v . The orbit number of a graph G , denoted by O r b ( G ) , is the number of orbits that partition V ( G ) . All vertex-transitive graphs G satisfy O r b ( G ) = 1 . Since the n -dimensional hypercube, denoted by Q n , is vertex-transitive, it follows that O r b ( Q n ) = 1 for n ≥ 1 . The twisted crossed cube, denoted by T C Q n , is a variant of the hypercube. In this paper, we prove that O r b ( T C Q n ) = 1 if n ≤ 4 , O r b ( T C Q 5 ) = O r b ( T C Q 6 ) = 2 , and O r b ( T C Q n ) = 2 ⌊ n − 1 2 ⌋ if n ≥ 7 .

Suggested Citation

  • Jia-Jie Liu, 2024. "The Orbits of Twisted Crossed Cubes," Mathematics, MDPI, vol. 12(24), pages 1-18, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:24:p:3928-:d:1543150
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