Symmetric property and the bijection between perfect matchings and sub-hypercubes of enhanced hypercubes

Abstract

An ( n , k ) -enhanced hypercube Q n , k ( 1 ≤ k ≤ n − 1 ) is an extended structure of an n -dimensional hypercube Q n . Q n , k possesses many properties that are superior to those of Q n including the diameter, fault diameter, and connectivity. For 1 ≤ i ≤ n , let P M i = { ( u , u i ) | u ∈ V ( Q n , k ) } , P M = { ( u , u ̄ ) | u ∈ V ( Q n , k ) } , where u = u 1 u 2 ⋯ u n , u ̄ = u 1 ⋯ u k − 1 u ̄ k u ̄ k + 1 ⋯ u ̄ n , and u i = u 1 u 2 ⋯ u i − 1 u ̄ i u i + 1 ⋯ u n , u i = 0 or 1, i = 1 , 2 , … , n . In this note, we prove that for any perfect matching M of Q n , k , if n ≥ 4 , 2 ≤ k ≤ n − 1 and k ≠ n − 2 or n = 3 , k = 2 , then Q n , k − M is isomorphic to Q n if and only if M = P M or P M i where i ∈ { k , k + 1 , … , n } , which extends Lü and Wu’s result in [Discrete Applied Mathematics, 279 (2020) 192–194]. Based on this result, we show that Q n , k with 3 ≤ k + 1 ≤ n is not edge-transitive.