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Hole: An Emerging Character in the Story of Radio k-Coloring Problem

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Mathematical and Statistical Applications in Life Sciences and Engineering

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Abstract

Frequency assignment problem (FAP) of radio networks is a very active area of research. In this chapter, radio networks have been studied in a graph-theoretic approach where the base stations of a cellular network are vertices and two vertices are adjacent if the corresponding stations transmit or receive broadcast of each other. Here, we deal with the problem of allocating frequencies to the base stations of the cellular network such that interference between stations at different distances can be avoided and at the same time, and the range of distinct frequencies used can be kept minimum. A certain variant of FAP on such network is radio k-coloring problem (\(k\ge 2\) being a positive integer) where stations (vertices) are assigned frequencies (colors) in such a way that the frequency difference increases (using k as a parameter) with the growing proximity of the stations. The focus has been laid on unused frequencies in the spectrum. These unused frequencies are referred as holes, and they are found to heavily influence the structure of the network (graph). In this chapter, we highlight many combinatorial aspects of holes in the context of radio k-coloring problem and its applicability as well as importance in real life.

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Acknowledgements

The research is supported in part by National Board for Higher Mathematics, Department of Atomic Energy, Government of India (No 2/48(10)/2013/NBHM(R.P.)/R&D II/695).

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Kolkata, 700019, India

    Ushnish Sarkar & Avishek Adhikari

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  1. Ushnish Sarkar
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  2. Avishek Adhikari
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Correspondence to Avishek Adhikari .

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Editors and Affiliations

  1. Department of Pure Mathematics, University of Calcutta, Kolkata, West Bengal, India

    Avishek Adhikari

  2. IMBIC, Kolkata, West Bengal, India

    Mahima Ranjan Adhikari

  3. Department of Mathematics and Statistics, Concordia University, Montreal, Québec, Canada

    Yogendra Prasad Chaubey

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Sarkar, U., Adhikari, A. (2017). Hole: An Emerging Character in the Story of Radio k-Coloring Problem. In: Adhikari, A., Adhikari, M., Chaubey, Y. (eds) Mathematical and Statistical Applications in Life Sciences and Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-10-5370-2_1

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