Sharp Bounds for Bandwidth of Clique Products

TL;DR

This paper establishes sharp lower bounds for the bandwidth of clique product graphs and provides an algorithm for optimal labeling, extending known results from hypercubes to arbitrary clique sizes.

Contribution

It generalizes the bandwidth bounds from hypercubes to products of arbitrary-sized cliques and offers a constructive algorithm for optimal labeling.

Findings

Derived tight lower bounds for clique product bandwidths

Developed an algorithm for optimal vertex labeling in clique products

Extended classical hypercube results to more general graph products

Abstract

The bandwidth of a graph is the labeling of vertices with minimum maximum edge difference. For many graph families this is NP-complete. A classic result computes the bandwidth for the hypercube. We generalize this result to give sharp lower bounds for products of cliques. This problem turns out to be equivalent to one in communication over multiple channels in which channels can fail and the information sent over those channels is lost. The goal is to create an encoding that minimizes the difference between the received and the original information while having as little redundancy as possible. Berger-Wolf and Reingold [2] have considered the problem for the equal size cliques (or equal capacity channels). This paper presents a tight lower bound and an algorithm for constructing the labeling for the product of any number of arbitrary size cliques.