# A General Approach to *L*(*h,k*)-Label Interconnection Networks

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## Abstract

Given two non-negative integers *h* and *k*, an *L*(*h*, *k*)-*labeling* of a graph *G* = (*V*, *E*) is a function from the set *V* to a set of colors, such that adjacent nodes take colors at distance at least *h*, and nodes at distance 2 take colors at distance at least *k*. The aim of the *L*(*h*, *k*)-labeling problem is to minimize the greatest used color. Since the decisional version of this problem is NP-complete, it is important to investigate particular classes of graphs for which the problem can be efficiently solved. It is well known that the most common interconnection topologies, such as Butterfly-like, Bene·s, CCC, Trivalent Cayley networks, are all characterized by a similar structure: they have nodes organized as a matrix and connections are divided into layers. So we naturally introduce a new class of graphs, called (*l* × *n*)-*multistage graphs*, containing the most common interconnection topologies, on which we study the *L*(*h*, *k*)-labeling. A general algorithm for *L*(*h*, *k*)-labeling these graphs is presented, and from this method an efficient *L*(2, 1)-labeling for Butterfly and CCC networks is derived. Finally we describe a possible generalization of our approach.

## Keywords

multistage interconnection network*L*(

*h*,

*k*)-labeling channel assignment problem

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