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

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.