The Orbits of Twisted Crossed Cubes

Jia-Jie Liu ()
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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

Mathematics, 2024, vol. 12, issue 24, 1-18

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 .

Keywords: twisted crossed cubes; automorphism; vertex-transitive; orbits (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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