On embedding complete graphs into hypercubes

AbstractAn embedding of Kn into a hypercube is a mapping, gf, of the n vertices of Kn to distinct vertices of the hypercube. The associated cost is the sum over all pairs of vertices, vi, vj, i⩽j, of the (Hamming) distance between gf(vi) and gf(vj). Let tf(n) denote the minimum cost over all embeddings of Kn into a hypercube (of any dimension). In this note we prove that tf(n) = (n − 1)2 unless n = 4 or 8, in which case tf(n) = (n − 1)2 − 1. As an application, we use this theorem to derive an alternate proof of the fact that the Isolation Heuristic (and its accompanying variants) for the multiway cut problem of Dahlhaus et al. (1994) are tight for all n. This result also gives a combinatorial justification for the seemingly anomalous improvements that these variants achieve in the cases n = 4 and 8

Embedding cube-connected cycles graphs into faulty hypercubes

We consider the problem of embedding a cube-connected cycles graph (CCC) into a hypercube with edge faults. Our main result is an algorithm that, given a list of faulty edges, computes an embedding of the CCC that spans all of the nodes and avoids all of the faulty edges. The algorithm has optimal running time and tolerates the maximum number of faults (in a worst-case setting). Because ascend-descend algorithms can be implemented efficiently on a CCC, this embedding enables the implementation of ascend-descend algorithms, such as bitonic sort, on hypercubes with edge faults. We also present a number of related results, including an algorithm for embedding a CCC into a hypercube with edge and node faults and an algorithm for embedding a spanning torus into a hypercube with edge faults

Boxicity and Cubicity of Product Graphs

The 'boxicity' ('cubicity') of a graph G is the minimum natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes (axis-parallel unit cubes) in $R^k$. In this article, we give estimates on the boxicity and the cubicity of Cartesian, strong and direct products of graphs in terms of invariants of the component graphs. In particular, we study the growth, as a function of $d$, of the boxicity and the cubicity of the $d$-th power of a graph with respect to the three products. Among others, we show a surprising result that the boxicity and the cubicity of the $d$-th Cartesian power of any given finite graph is in $O(\log d / \log\log d)$ and $\theta(d / \log d)$, respectively. On the other hand, we show that there cannot exist any sublinear bound on the growth of the boxicity of powers of a general graph with respect to strong and direct products.Comment: 14 page