Hamiltonicity of the basic WK-recursive pyramid with and without faulty nodes

In 1993, Fernandes and Kanevsky proposed an important structure for interconnection networks, WKR Pyramid Networks (WKP(d,t,L)WKP(d,t,L), for short). These are constructed by taking difference size WK-recursive network (WK(d,tn)WK(d,tn), for short) as difference layers. That study discussed the orders, sizes and connectivity of WKP(d,t,L)WKP(d,t,L) for any integers d≥1,t≥1d≥1,t≥1 and L≥1L≥1. The basic WK-recursive pyramid, denoted by WKP(d,L)WKP(d,L), is a basic version of WKP(d,t,L)WKP(d,t,L) such that t=1t=1. In WKP(d,L)WKP(d,L), each vertex has exactly d children and the n  th layer is isomorphic to a WK(d,n)WK(d,n). In this paper, we show that WKP(d,L)WKP(d,L) is Hamiltonian-connected for any two integers, d≥3d≥3 and L≥1L≥1, and it is also (d−2)(d−2)-node Hamiltonian for any two integers, d≥2d≥2 and L≥1L≥1.