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

Y1 - 2010

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.

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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

T2 - 21st Annual International Symposium on Algorithms and Computations, ISAAC 2010

Y2 - 14 December 2010 through 16 December 2010

ER -