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

Toru Hasunuma, Toshimasa Ishii, Hirotaka Ono, Yushi Uno

7 被引用数 (Scopus)

## 抄録

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.

本文言語 英語 1407-1420 14 Discrete Mathematics 312 8 https://doi.org/10.1016/j.disc.2012.01.007 出版済み - 4 28 2012

## All Science Journal Classification (ASJC) codes

• 理論的コンピュータサイエンス
• 離散数学と組合せ数学

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