### 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 λp,qT(G), among all possible assignments. In this paper, we first give new upper and lower bounds on λp,qT(G) for some classes of graphs G, in particular, tight bounds on λp,qT(T) for trees T. We then show that if p≤3q2, the problem for trees T is linearly solvable, and completely determine λp,qT(T) for trees T with Δ<4, 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 |
---|---|

Pages (from-to) | 1407-1420 |

Number of pages | 14 |

Journal | Discrete Mathematics |

Volume | 312 |

Issue number | 8 |

DOIs | |

Publication status | Published - Apr 28 2012 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*312*(8), 1407-1420. https://doi.org/10.1016/j.disc.2012.01.007

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

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 312, no. 8, pp. 1407-1420. https://doi.org/10.1016/j.disc.2012.01.007

}

TY - JOUR

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

AU - Hasunuma, Toru

AU - Ishii, Toshimasa

AU - Ono, Hirotaka

AU - Uno, Yushi

PY - 2012/4/28

Y1 - 2012/4/28

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)|

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)|

UR - http://www.scopus.com/inward/record.url?scp=84856304549&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856304549&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2012.01.007

DO - 10.1016/j.disc.2012.01.007

M3 - Article

AN - SCOPUS:84856304549

VL - 312

SP - 1407

EP - 1420

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 8

ER -