Abstract
In this paper, we address a particular case of the general problem of λ 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 λ number of G, and is denoted by λ \(_{q}^{p}\) (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 no-holeL(p,q) labelling problem for G. Let [0;M′] be this new span and call the ν number of G the value M′, and denote it by ν \(^{p}_{q}\)(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 ν \(_{\rm 0}^{p}\) 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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bertossi, A.A., Pinotti, C.M., Tan, R.B.: Efficient use of radio spectrum in wireless networks with channel separation between close stations. In: Proc. DIAL M for Mobility 2000 (Fourth International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications) (2000)
Bertossi, A.A., Pinotti, C.M., Tan, R.B.: Channel assignment with separation for special classes of wireless networks: Grids and rings. In: Proc. IPDPS 2002 (International Parallel and Distributed Processing Symposium), pp. 28–33. IEEE Computer Society, Los Alamitos (2002)
Bodlaender, H.L., Kloks, T., Tan, R.B., van Leeuwen, J.: λ-coloring of graphs. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 395–406. Springer, Heidelberg (2000)
Calamoneri, T., Petreschi, R.: L(2, 1)-labeling of planar graphs. In: Proc. DIAL M for Mobility 2001 (Fifth International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications), pp. 28–33 (2001)
Chang, G.J., Ke, W.-T., Kuo, D., Liu, D.D.-F., Yeh, R.K.: On L(d1)-labelings of graphs. Discrete Mathematics 220, 57–66 (2002)
Chang, G.J., Kuo, D.: The L(2, 1)-labeling on graphs. SIAM J. Discrete Math. 9, 309–316 (1996)
Fertin, G., Raspaud, R.: L(p,q) labeling of d-dimensional grids. Discrete Mathematics (2003)
Griggs, J.R., Yeh, R.K.: Labeling graphs with a condition at distance two. SIAM J. Discrete Math. 5, 586–595 (1992)
Hale, W.K.: Frequency assignment: theory and applications. Proc. IEEE 60, 1497–1514 (1980)
Jha, P.K.: Optimal L(2, 1)-labeling of cartesian products of cycles with an application to independent domination. IEEE Trans. Circuits & Systems I: Fundamental Theory and Appl. 47, 1531–1534 (2000)
Jha, P.K., Narayanan, A., Sood, P., Sundaram, K., Sunder, V.: On L(2, 1)-labeling of the cartesian product of a cycle and a path. Ars Combin. 55, 81–89 (2000)
Metzger, B.H.: Spectrum management technique. In: Paper presented at the 38th National ORSA Meeting, Detroit, MI (1970)
Molloy, M., Salavatipour, M.R.: Frequency channel assignment on planar networks. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 736–747. Springer, Heidelberg (2002)
Sakai, D.: Labeling chordal graphs with a condition at distance two. SIAM J. Discrete Math. 7, 133–140 (1994)
van den Heuvel, J.: Radio channel assignment on 2-dimensional lattices. Annals of Combinatorics 6, 463–477 (2002)
van den Heuvel, J., Leese, R.A., Shepherd, M.A.: Graph labelling and radio channel assignment. Journal of Graph Theory 29, 263–284 (1998)
Whittlesey, M., Georges, J., Mauro, D.W.: On the λ-number of Qn and related graphs. SIAM J. Discrete Math. 8, 499–506 (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fertin, G., Raspaud, A., Sýkora, O. (2004). No-Hole L(p,0) Labelling of Cycles, Grids and Hypercubes. In: Královic̆, R., Sýkora, O. (eds) Structural Information and Communication Complexity. SIROCCO 2004. Lecture Notes in Computer Science, vol 3104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27796-5_13
Download citation
DOI: https://doi.org/10.1007/978-3-540-27796-5_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22230-9
Online ISBN: 978-3-540-27796-5
eBook Packages: Springer Book Archive