A Linear Time Algorithm for L(2,1)-Labeling of Trees

Toru Hasunuma, Toshimasa Ishii, Hirotaka Ono, Yushi Uno

Research output: Chapter in Book/Report/Conference proceedingConference contribution

19 Citations (Scopus)


An L(2,1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that |f(x)-f(y)|≥2 if x and y are adjacent and |f(x)-f(y)|≥1 if x and y are at distance 2, for all x and y in V(G). A k-L(2,1)-labeling is an L(2,1)-labeling f:V(G)→{0,...,k}, and the L(2,1)-labeling problem asks the minimum k, which we denote by λ(G), among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2, and tree is one of very few classes for which the problem is polynomially solvable. The running time of the best known algorithm for trees had been O(Δ4.5 n) for more than a decade, and an O( min {n 1.751.5 n})-time algorithm has appeared recently, where Δ is the maximum degree of T and n=|V(T)|, however, it has been open if it is solvable in linear time. In this paper, we finally settle this problem for L(2,1)-labeling of trees by establishing a linear time algorithm.

Original languageEnglish
Title of host publicationAlgorithms - ESA 2009 - 17th Annual European Symposium, Proceedings
Number of pages12
Publication statusPublished - 2009
Event17th Annual European Symposium on Algorithms, ESA 2009 - Copenhagen, Denmark
Duration: Sept 7 2009Sept 9 2009

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5757 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other17th Annual European Symposium on Algorithms, ESA 2009

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)


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