### Abstract

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 _{pq}^{T}(G), among all possible assignments. In this paper, we first give new upper and lower bounds on λ_{pq}^{T}(G) for some classes of graphs G, in particular, tight bounds on λ_{pq}^{T}(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 λ_{pq}^{T}(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 language | English |
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Title of host publication | Algorithms and Computation - 21st International Symposium, ISAAC 2010, Proceedings |

Pages | 49-60 |

Number of pages | 12 |

Edition | PART 2 |

DOIs | |

Publication status | Published - Dec 1 2010 |

Event | 21st Annual International Symposium on Algorithms and Computations, ISAAC 2010 - Jeju Island, Korea, Republic of Duration: Dec 15 2010 → Dec 17 2010 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 2 |

Volume | 6507 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 21st Annual International Symposium on Algorithms and Computations, ISAAC 2010 |
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Country | Korea, Republic of |

City | Jeju Island |

Period | 12/15/10 → 12/17/10 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Algorithms and Computation - 21st International Symposium, ISAAC 2010, Proceedings*(PART 2 ed., pp. 49-60). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6507 LNCS, No. PART 2). https://doi.org/10.1007/978-3-642-17514-5_5