Labelling graphs with a condition at distance 2
SIAM Journal on Discrete Mathematics
On the size of graphs labeled with condition at distance two
Journal of Graph Theory
The $L(2,1)$-Labeling Problem on Graphs
SIAM Journal on Discrete Mathematics
Coloring precolored perfect graphs
Journal of Graph Theory
Assigning codes in wireless networks: bounds and scaling properties
Wireless Networks
Fixed-Parameter Complexity of lambda-Labelings
WG '99 Proceedings of the 25th International Workshop on Graph-Theoretic Concepts in Computer Science
Graph labeling and radio channel assignment
Journal of Graph Theory
Radio Labeling with Pre-assigned Frequencies
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Systems of distant representatives
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
List version of L(d, s)-labelings
Theoretical Computer Science - Graph colorings
Systems of distant representatives
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
Distance constrained labelings of trees
TAMC'08 Proceedings of the 5th international conference on Theory and applications of models of computation
The L(h,1,1)-labelling problem for trees
European Journal of Combinatorics
Distance constrained labelings of graphs of bounded treewidth
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
L(2, 1, 1)-labeling is NP-complete for trees
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Elegant distance constrained labelings of trees
WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
Distance three labelings of trees
Discrete Applied Mathematics
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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Graph colorings with distance constraints are motivated by the frequency assignment problem. The so called 驴(p,q)-labeling problem asks for coloring the vertices of a given graph with integers from the range {0, 1, ..., 驴} so that labels of adjacent vertices differ by at least p and labels of vertices at distance 2 differ by at least q, where p, q are fixed integers and integer 驴 is part of the input. It is known that this problem is NP-complete for general graphs, even when 驴 is fixed, i.e., not part of the input, but polynomially solvable for trees for (p,q)=(2,1). It was conjectured that the general case is also polynomial for trees. We consider the precoloring extension version of the problem (i.e., when some vertices of the input tree are already precolored) and show that in this setting the cases q=1 and q 1 behave differently: the problem is polynomial for q=1 and any p, and it is NP-complete for any p q 1.