A linear algorithm for radio \(k\)-coloring of powers of paths having small diameters (Q6627041)

The following questions are relevant to the paper:\NQ1. The Channel Assignment Problem (CAP) in wireless networks can be modeled using different graph labeling problems with distance constraints. Why do distance constraints happen? Q2. The frequency separation varies inversely proportional to the distance, which is a correct requirement for a real-life model. Is there an example? Q3. Undirected simple graphs are considered here. Is there any reason for choosing such a graph? Q4. What is the optimum value of \(k\) for the proposed method? Q5. The layer \(L_i\) is defined as the set of vertices at a distance \(i\) from \(L_0\). Specify the maximum value of \(i\). A theorem relates to this. Q6. How much iteration must be done to find the upper bound of a vertex?\N\NReviewer's remarks: A considerable amount of attention has been given to finding the exact values or providing polynomial time algorithms to determine \(rck(G)\) for several basic graph families such as paths, cycles, trees, and powers of paths, usually for some specific values of \(k\). The authors find the exact values of \(rck(G)\) where \(G\) is the power of a path with a diameter strictly less than \(k\). The proof readily provides a linear time algorithm for assigning a radio \(k\)-coloring of \(G\). Notice that, finding the exact value of \(rck(G)\), or providing a polynomial time algorithm to determine it for so-called well-understood families of graphs, such as paths, cycles, trees, and powers of paths, seems to be difficult and challenging. The graph \(P^m_n\) is obtained by adding edges between the vertices of \(P_n\) that are at most \(m\) distance apart. Notice that, the so-obtained graph is, in particular, an interval graph. In this article, the authors develop a robust graph theoretic tool for the proof. Even though the tool is specifically used to prove this, it can be adapted to prove bounds for other classes of graphs. Thus, the main contribution of this work is not only in proving an important result that captures a significant number of problems with a unified proof but also in devising a proof technique that has the potential to become a standard technique to attack similar problems. The proof of the upper bound for Theorem 1.8 is given by prescribing an algorithm (implicitly). The algorithm requires ordering the vertices of the input graph, and then providing the coloring based on the ordering. Each step runs in linear order of the number of vertices in the input graph. Moreover, the authors prove theoretically the tightness of the upper bound. The work is excellent but it does not describe any practical example.