Labelling graphs with a condition at distance 2
SIAM Journal on Discrete Mathematics
The traveling salesman problem with distances one and two
Mathematics of Operations Research
The $L(2,1)$-Labeling Problem on Graphs
SIAM Journal on Discrete Mathematics
Distance Constrained Labeling of Precolored Trees
ICTCS '01 Proceedings of the 7th Italian Conference on Theoretical Computer Science
A Hamiltonian Approach to the Assignment of Non-reusable Frequencies
Proceedings of the 18th Conference on Foundations of Software Technology and Theoretical Computer Science
An explicit lower bound for TSP with distances one and two
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
On algorithms for (P5,gem)-free graphs
Theoretical Computer Science - Graph colorings
A general framework for coloring problems: old results, new results, and open problems
IJCCGGT'03 Proceedings of the 2003 Indonesia-Japan joint conference on Combinatorial Geometry and Graph Theory
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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 pre-assigned labels and an arbitrary number of pre-assigned 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 驴 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 驴 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.