Embedding hamiltonian paths in hypercubes with a required vertex in a fixed position

Abstract

Assume that n is a positive integer with n ⩾ 2 . It is proved that between any two different vertices x and y of Q n there exists a path P l ( x , y ) of length l for any l with h ( x , y ) ⩽ l ⩽ 2 n − 1 and 2 | ( l − h ( x , y ) ) . We expect such path P l ( x , y ) can be further extended by including the vertices not in P l ( x , y ) into a hamiltonian path from x to a fixed vertex z or a hamiltonian cycle. In this paper, we prove that for any two vertices x and z from different partite set of n-dimensional hypercube Q n , for any vertex y ∈ V ( Q n ) − { x , z } , and for any integer l with h ( x , y ) ⩽ l ⩽ 2 n − 1 − h ( y , z ) and 2 | ( l − h ( x , y ) ) , there exists a hamiltonian path R ( x , y , z ; l ) from x to z such that d R ( x , y , z ; l ) ( x , y ) = l . Moreover, for any two distinct vertices x and y of Q n and for any integer l with h ( x , y ) ⩽ l ⩽ 2 n − 1 and 2 | ( l − h ( x , y ) ) , there exists a hamiltonian cycle S ( x , y ; l ) such that d S ( x , y ; l ) ( x , y ) = l .