Abstract
Graph colorings with distance constraints are motivated by the frequency assignment problem. The so called λ(p,q)-labeling problem asks for coloring the vertices of a given graph with integers from the range {0, 1, ..., λ} 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 λ is part of the input. It is known that this problem is NP-complete for general graphs, even when λ 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 di.erently: the problem is polynomial for q=1 and any p, and it is NP-complete for any p > q > 1.
The authors acknowledge support of joint Czech U.S. grants KONTAKT ME338 and NSF-INT-9802416 during visits of the first two authors to Eugene, OR, and of the third author to Prague.
Partially supported by EU ARACNE project HPRN-CT-1999-00112. Supported by the Ministery of Education of the Czech Republic as project LN00A056
Research partially supported by Czech Research grant GAUK 158/99. Supported by the Ministery of Education of the Czech Republic as project LN00A056
Supported in part by the grant NSF-ANI-9977524.
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Fiala, J., Kratochvíl, J., Proskurowski, A. (2001). Distance Constrained Labeling of Precolored Trees. In: Theoretical Computer Science. ICTCS 2001. Lecture Notes in Computer Science, vol 2202. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45446-2_18
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DOI: https://doi.org/10.1007/3-540-45446-2_18
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