The (p,q)-total labeling problem for trees

Toru Hasunuma, Toshimasa Ishii, Hirotaka Ono, Yushi Uno

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


A (p,q)-total labeling of a graph G is an assignment f from the vertex set V(G) and the edge set E(G) to the set of nonnegative integers such that |f(x)-f(y)| ≥ p if x is a vertex and y is an edge incident to x, and |f(x) - f(y)| ≥ q if x and y are a pair of adjacent vertices or a pair of adjacent edges, for all x and y in V(G) ∪ E(G). A k-(p,q)-total labeling is a (p,q)-total labeling f:V(G) ∪ E(G)→{0,...,k}, and the (p,q)-total labeling problem asks the minimum k, which we denote by λ pqT(G), among all possible assignments. In this paper, we first give new upper and lower bounds on λpqT(G) for some classes of graphs G, in particular, tight bounds on λpqT(T) for trees T. We then show that if p ≤ 3q/2, the problem for trees T is linearly solvable, and give a complete characterization of trees achieving λpqT(T) if in addition Δ ≥ 4 holds, where Δ is the maximum degree of T. It is contrasting to the fact that the L(p,q)-labeling problem, which is a generalization of the (p,q)-total labeling problem, is NP-hard for any two positive integers p and q such that q is not a divisor of p.

Original languageEnglish
Title of host publicationAlgorithms and Computation - 21st International Symposium, ISAAC 2010, Proceedings
Number of pages12
EditionPART 2
Publication statusPublished - 2010
Event21st Annual International Symposium on Algorithms and Computations, ISAAC 2010 - Jeju Island, Korea, Republic of
Duration: Dec 15 2010Dec 17 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 2
Volume6507 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other21st Annual International Symposium on Algorithms and Computations, ISAAC 2010
Country/TerritoryKorea, Republic of
CityJeju Island

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)


Dive into the research topics of 'The (p,q)-total labeling problem for trees'. Together they form a unique fingerprint.

Cite this