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

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 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. © 2010 Springer-Verlag.