Labelling graphs with a condition at distance 2
SIAM Journal on Discrete Mathematics
Regular codes in regular graphs are difficult
Discrete Mathematics
The $L(2,1)$-Labeling Problem on Graphs
SIAM Journal on Discrete Mathematics
Journal of Combinatorial Theory Series B
Complexity of graph covering problems
Nordic Journal of Computing
Complexity of Colored Graph Covers I. Colored Directed Multigraphs
WG '97 Proceedings of the 23rd International Workshop on Graph-Theoretic Concepts in Computer Science
IWDC '02 Proceedings of the 4th International Workshop on Distributed Computing, Mobile and Wireless Computing
Distance Constrained Labeling of Precolored Trees
ICTCS '01 Proceedings of the 7th Italian Conference on Theoretical Computer Science
L(2, 1)-Coloring Matrogenic Graphs
LATIN '02 Proceedings of the 5th Latin American Symposium on Theoretical Informatics
Online and Offline Distance Constrained Labeling of Disk Graphs
ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
Note: L(2,1)-Labelings on the composition of n graphs
Theoretical Computer Science
TAPAS'11 Proceedings of the First international ICST conference on Theory and practice of algorithms in (computer) systems
The L(2,1)-labeling of unigraphs
Discrete Applied Mathematics
Generalized powers of graphs and their algorithmic use
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
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A λ-labeling of a graph G is an assignment of labels from the set {0, . . . , λ} to the vertices of a graph G such that vertices at distance at most two get different labels and adjacent vertices get labels which are at least two apart. We study the minimum value λ = λ(G) such that G admits a λ-labeling. We show that for every fixed value k ≥ 4 it is NP-complete to determine whether λ(G) ≤ k. We further investigate this problem for sparse graphs (k-almost trees), extending the already known result for ordinary trees. In a generalization of this problem we wish to find a labeling such that vertices at distance two are assigned labels that differ by at least q and the labels of adjacent vertices differ by at least p (where p and q are given positive integers). We denote the minimum number of labels by L(G; p, q) (hence λ(G) = L(G; 2, 1)). We show several hardness results for L(G; p, q) including that for any p q ≥ 1 there is a λ = λ(p, q) such that deciding if L(G; p, q) ≤ λ(p, q) is NP-complete.