Radio labeling with preassigned frequencies

Abstract

A radio labeling of a graph G is an assignment of pairwise distinct, positive integer labels to the vertices of G such that labels of adjacent vertices differ by at least $2$. The radio labeling problem (RL) consists in determining a radio labeling that minimizes the maximum label that is used (the so-called span of the labeling). RL is a well-studied problem, mainly motivated by frequency assignment problems in which transmitters are not allowed to operate on the same frequency channel. We consider the special case where some of the transmitters have preassigned operating frequency channels. This leads to the natural variants p-RL(l) and p-RL(*) of RL with l preassigned labels and an arbitrary number of preassigned labels, respectively. We establish a number of combinatorial, algorithmical, and complexity-theoretical results for these variants of radio labeling. In particular, we investigate a simple upper bound on the minimum span, yielding a linear time approximation algorithm with a constant additive error bound for p-RL(*) restricted to graphs with girth at least 5. We consider the complexity of p-RL(l) and p-RL(*) for several cases in which RL is known to be polynomially solvable. On the negative side, we prove that p-RL(*) is NP-hard for cographs and for k-colorable graphs where a k-coloring is given (k at least 3). On the positive side, we derive polynomial time algorithms solving p-RL(*)} and p-RL(l) for graphs with bounded maximum degree, and for solving p-RL(l) for k-colorable graphs where a k-coloring is given.