LINEAR LAYOUT OF GENERALIZED HYPERCUBES

Abstract

This paper deals with two kinds of generalized hypercubes: a d-dimensional c-ary clique [Formula: see text] and a d-dimensional c-ary array [Formula: see text]. A d-dimensional c-ary clique [Formula: see text] has nodes labeled by cdintegers from 0 to cd- 1 and two nodes are connected by an edge if and only if the c-ary representations of their labels differ by one and only one digit. A d-dimensional c-ary array [Formula: see text] also has nodes labeled by cdintegers from 0 to cd- 1, and two nodes are connected if and only if the c-ary representations of their labels differ by one and only one digit and the absolute value of the difference in that digit is 1. Further, an n-node c-ary clique [Formula: see text] is the induced subgraph of [Formula: see text] with nodes labeled by integers from 0 to n - 1. The main contribution of this paper is to clarify several topological properties of [Formula: see text] and [Formula: see text] in terms of their linear layouts. For this purpose, we first prove that [Formula: see text] is a maximum subgraph of [Formula: see text], that is, [Formula: see text]has the largest number of edges over all n-node subgraphs of [Formula: see text], whenever n ≤ m. Using this fact, we show the exact values of the bisection width, cut width, and total edge length of [Formula: see text]. We also show the exact value of the bisection width of [Formula: see text] and nearly tight values of the cut width and the total edge length of [Formula: see text].