### 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 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### 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

**The (p,q)-total labeling problem for trees.** / Hasunuma, Toru; Ishii, Toshimasa; Ono, Hirotaka; Uno, Yushi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithms and Computation - 21st International Symposium, ISAAC 2010, Proceedings.*PART 2 edn, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), no. PART 2, vol. 6507 LNCS, pp. 49-60, 21st Annual International Symposium on Algorithms and Computations, ISAAC 2010, Jeju Island, Korea, Republic of, 12/15/10. https://doi.org/10.1007/978-3-642-17514-5_5

}

TY - GEN

T1 - The (p,q)-total labeling problem for trees

AU - Hasunuma, Toru

AU - Ishii, Toshimasa

AU - Ono, Hirotaka

AU - Uno, Yushi

PY - 2010/12/1

Y1 - 2010/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=78650861820&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650861820&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-17514-5_5

DO - 10.1007/978-3-642-17514-5_5

M3 - Conference contribution

AN - SCOPUS:78650861820

SN - 3642175163

SN - 9783642175169

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 49

EP - 60

BT - Algorithms and Computation - 21st International Symposium, ISAAC 2010, Proceedings

ER -