On the S-Labeling problem

Let G be a graph of order n and size m. A labeling of G is a bijective mapping θ : V(G) → {1, 2, . . . , n}, and we call Θ(G) the set of all labelings of G. For any graph G and any labeling θ ∈ Θ(G), let SL(G, θ) = ∑ e∈E(G)min{θ(u) : u ∈ e}. In this paper, we consider the S-Labeling problem, defined as follows: Given a graph G, find a labeling θ ∈ Θ(G) that minimizes SL(G, θ). The S-Labeling problem has been shown to be NP-complete [Via06]. We prove here basic properties of any optimal S-labeling of a graph G, and relate it to the Vertex Cover problem. Then, we derive bounds for SL(G, θ), and we give approximation ratios for different families of graphs. We finally show that the S-Labeling problem is polynomialtime solvable for split graphs. Due to space constraints, proofs are totally absent from this paper. They will be available in its journal version. 1 Preliminaries and Basic Properties We assume readers have basic knowledge about graph theory [Die00] and we shall thus use most conventional terms of graph theory without defining them (we will only recall basic notations here). Let G be a graph. We write V(G) for the set of vertices, E(G) for the set of edges, and ∆(G) (or ∆, if it is clear from the context) for the maximum degree of G. The order (resp. size) of G is ha l-0 04 16 57 0, v er si on 1 15 S ep 2 00 9 Author manuscript, published in "Proc. 5th Euroconference on Combinatorics, Graph Theory and Applications (EUROCOMB 2009), Bordeaux : France (2009)"