No-Hole L(p,0)-Labelling of Cycles, Grids and Hypercubes
Résumé
In this paper, we address a particular case of the general problem of $\lambda$ labellings, concerning frequency assignment for telecommunication networks. In this model, stations within a given radius $r$ must use frequencies that differ at least by a value $p$, while stations that are within a larger radius $r'>r$ must use frequencies that differ by at least another value $q$. The aim is to minimize the span of frequencies used in the network. This can be modelled by a graph labelling problem, called the $L(p,q)$ labelling, where one wants to label vertices of the graph $G$ modelling the network by integers in the range $[0;M]$, while minimizing the value of $M$. $M$ is then called the $\lambda$ number of $G$, and is denoted by $\lambda_q^p (G)$. Another parameter that sometimes needs to be optimized is the fact that all the possible frequencies (i.e., all the possible values in the span) are used. In this paper, we focus on this problem. More precisely, we want that: (1) all the frequencies are used and (2) condition~(1) being satisfied, the span must be minimum. We call this the {\em no-hole} $L(p,q)$ labelling problem for $G$. Let $[0;M']$ be this new span and call the $\nu$ number of $G$ the value $M'$, and denote it by $\nu^p_q(G)$. In this paper, we study a special case of no-hole $L(p,q)$ labelling, namely where $q=0$. We also focus on some specific topologies: cycles, hypercubes, 2-dimensional grids and 2-dimensional tori. For each of the mentioned topologies cited above, we give bounds on the $\nu_0^p$ number and show optimality in some cases. The paper is concluded by giving new results concerning the (general, i.e. not necessarily no-hole) $L(p,q)$ labelling of hypercubes.
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