Some Hamiltonian results in powers of graphs

Abstract

In this paper we show that the connectivity of the kth power of a graph of connectivity m is at least km if the kth power of the graph is not a complete graph. Also, we. prove th at removing as many as k -2 vertices from the kth power of a graph (k ;;. 3) leaves a Hamiltonian graph, and that removing as many as k -3 vertices from the kth power of a graph (k;;' 3) leaves a Hamiltonian con nected graph. Further, if every vertex of a graph has degree two or more, then the square of th e graph contai ns a 2-factor. Finally, we show that the squares of certain Euler graphs are Hamiltonian.