# Channel assignment problem and *n*-fold *t*-separated \(L(j_1,j_2,\ldots ,j_m)\)-labeling of graphs

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## Abstract

This paper considers the channel assignment problem in mobile communications systems. Suppose there are many base stations in an area, each of which demands a number of channels to transmit signals. The channels assigned to the same base station must be separated in some extension, and two channels assigned to two different stations that are within a distance must be separated in some other extension according to the distance between the two stations. The aim is to assign channels to stations so that the interference is controlled within an acceptable level and the spectrum of channels used is minimized. This channel assignment problem can be modeled as the multiple *t*-separated \(L(j_1,j_2,\ldots ,j_m)\)-labeling of the interference graph. In this paper, we consider the case when all base stations demand the same number of channels. This case is referred as *n*-fold *t*-separated \(L(j_1,j_2,\ldots ,j_m)\)-labeling of a graph. This paper first investigates the basic properties of *n*-fold *t*-separated \(L(j_1,j_2,\ldots ,j_m)\)-labelings of graphs. And then it focuses on the special case when \(m=1\). The optimal *n*-fold *t*-separated *L*(*j*)-labelings of all complete graphs and almost all cycles are constructed. As a consequence, the optimal *n*-fold *t*-separated \(L(j_1,j_2,\ldots ,j_m)\)-labelings of the triangular lattice and the square lattice are obtained for the case \(j_1=j_2=\cdots =j_m\). This provides an optimal solution to the corresponding channel assignment problems with interference graphs being the triangular lattice and the square lattice, in which each base station demands a set of *n* channels that are *t*-separated and channels from two different stations at distance at most *m* must be \(j_1\)-separated. We also study a variation of *n*-fold *t*-separated \(L(j_1,j_2,\ldots ,j_m)\)-labeling, namely, *n*-fold *t*-separated consecutive \(L(j_1,j_2,\ldots ,j_m)\)-labeling. And present the optimal *n*-fold *t*-separated consecutive *L*(*j*)-labelings of all complete graphs and cycles.

## Keywords

Channel assignment problem \(L(j_1, j_2, \ldots , j_m)\)-labeling number*n*-fold \(L(j_1, j_2, \ldots , j_m)\)-labeling number

*n*-fold

*t*-separated consecutive \(L(j_1, j_2, \ldots , j_m)\)-labeling number Complete graph Odd cycle Triangular lattice Square lattice

## Notes

### Acknowledgements

The authors wish to thank the reviewers for many valuable suggestions.

## References

- Audhya GK, Sinha K, Ghosh SC, Sinha BP (2011) A survey on the channel assignment problem in wireless networks. Wirel Commun Mob Comput 11:583–609CrossRefGoogle Scholar
- Calamoneri T (2011) The L(h, k)-labelling problem: an updated survey and annotated bibliography. Comput J 54(8):1344–1371CrossRefGoogle Scholar
- Campêloa M, Corrêa RC, Mourac PFS, Santos MC (2013) On optimal \(k\)-fold colorings of webs and antiwebs. Discrete Appl Math 161:60–70MathSciNetCrossRefGoogle Scholar
- Chvátal V, Garey MR, Johnson DS (1978) Two results concerning multicoloring. Ann Discrete Math 2:151–154MathSciNetCrossRefMATHGoogle Scholar
- Díaz IM, Zabala P (1999) A generalization of the graph coloring problem. Investig Oper 8(1–3):167–184Google Scholar
- Ghosh SC, Sinha BP, Das N (2003) Channel assignment using genetic algorithm based on geometric. IEEE Trans Veh Technol 52(4):860–875CrossRefGoogle Scholar
- Griggs JR, Jin XT (2007) Recent progress in mathematics and engineering on optimal graph labellings with distance conditions. J Comb Optim 14(2–3):249–257MathSciNetCrossRefMATHGoogle Scholar
- Griggs JR, Yeh RK (1992) Labeling graphs with a condition at distance 2. SIAM J Discrete Math 5:586–595MathSciNetCrossRefMATHGoogle Scholar
- Hale WK (1980) Frequency assignment: theory and applications. Proc IEEE 68:1497–1514CrossRefGoogle Scholar
- Janssen J (2002) Channel assignment and graph labeling. In: Handbook of wireless networks and mobile computing. John Wiley & Sons, Inc., pp 95–117Google Scholar
- Johnson A, Holroyd FC, Stahl S (1997) Multichromatic numbers, star chromatic numbers and Kneser graphs. J Graph Theory 26:137–145MathSciNetCrossRefMATHGoogle Scholar
- Khanna S, Kumaran K (1998) On wireless spectrum estimation and generalized graph coloring. In: Proceedings of IEEE INFOCOM98,Google Scholar
- Klostermeyer W, Zhang CQ (2002) \(n\)-tuple coloring of planar graphs with large odd girth. Graphs and Comb 18:119–132MathSciNetCrossRefMATHGoogle Scholar
- Lin W (2008) Multicolouring and Mycielski construction. Discrete Math 308:3565–3573MathSciNetCrossRefMATHGoogle Scholar
- Lin W, Zhang P (2012) On \(n\)-fold \(L(j, k)\)- and circular \(L(j, k)\)-labeling of graphs. Discrete Appl Math 160(16–17):2452–2461MathSciNetCrossRefMATHGoogle Scholar
- Lin W, Liu DDF, Zhu X (2010) Multi-colouring the Mycielskian of graphs. J Graph Theory 63(4):311–323MathSciNetMATHGoogle Scholar
- Pan Z, Zhu X (2010) Mutiple coloring of cone graphs. SIAM J Discrete Math 24(4):1515–1526MathSciNetCrossRefMATHGoogle Scholar
- Ren G, Bu Y (2010) \(k\)-fold coloring of planar graphs. Sci China Math 53(10):2791–2800MathSciNetCrossRefMATHGoogle Scholar
- Stahl S (1976) \(n\)-tuple colorings and associated graphs. J Comb Theory Ser B 20:185–203MathSciNetCrossRefMATHGoogle Scholar
- Thevenin S, Zufferey N, Potvin J-Y (2016) Graph multi-coloring for a job scheduling application. Discrete Appl Math. https://doi.org/10.1016/j.dam.2016.05.023 MATHGoogle Scholar
- van den Heuvel J, Leese RA, Shepherd MA (1998) Graph labeling and radio channel assignment. J Graph Theory 29:263–283MathSciNetCrossRefMATHGoogle Scholar
- Yeh RK (2006) A survey on labeling graphs with a condition at distance two. Discrete Math 306:1217–1231MathSciNetCrossRefMATHGoogle Scholar
- Zhang P, Lin W (2014) Multiple \(L(j,1)\)-labeling of the triangular lattice. J Comb Optim 27(4):695–710MathSciNetCrossRefMATHGoogle Scholar