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

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.