A Hypercube Variant with Small Diameter

Abstract

This article introduces a new variant of hypercubes Hn. The n‐dimensional twisted hypercube Hn is obtained from two copies of the (n−1)‐dimensional twisted hypercube Hn−1 by adding a perfect matching between the vertices of these two copies of Hn−1. We prove that the n‐dimensional twisted hypercube Hn has diameter false(1+o(1)false)n/prefixlog2n. This improves on the previous known variants of hypercube of dimension n and is optimal up to an error of order o(n/prefixlog2n). Another type of hypercube variant Zn,k that has similar structure and properties as Hn is also discussed in the last section.

A note on minimum linear arrangement for BC graphs

“…Fan et al [12] proposed a family of interconnection networks called BC graphs. BC networks are a class of networks which include several well-known interconnection networks like hypercubes, Möbius cubes [16], crossed cubes [17], twisted cubes [18], locally twisted cube [19], spined cube [20] and Z-cube [21]. These variations of hypercubes generally possess certain superior properties over the hypercubes and are recognized as attractive alternatives to the hypercubes.…”

Spanning trails avoiding and containing given edges

“…If we do not emphasize the sequence A, we use G i for G i (A). For example, the n-th iterated permutation graph of K 1 is a hypercube variant, a twisted hypercube of dimension n (see Definition 1.1 in [26]). By the definition of iterated permutation graphs, as well as Theorem 2.3 and Observation 2.4, we obtain the following observation.…”

A novel view: edge isoperimetric methods and reliability evaluation of several kinds of conditional edge-connectivity of interconnection networks

“…The n-dimensional bijective connection networks B n (also called hypercube-like networks) are a class of cube-based networks including several well known interconnection networks like hypercubes (denoted by Q n ), twisted cubes [9] (denoted by T Q n ), crossed cubes [5] (denoted by CQ n ), spined cubes SQ n , parity cubes P Q n , Z-cubes ZQ n [33], varietal cubes V Q n , Möbius cubes [4] (denoted by M Q n ), locally twisted cubes [27] (denoted by LT Q n ), restricted hypercube-like networks RHLN n , generalized cubes GQ n , generalized twisted cubes [1] (denoted by GT Q n ) and Mcubes [24] (denoted by M CQ n ) as members.…”