On the L(p, 1)-labelling of graphs

In all the paper we work on a graphG with maximal degree∆. For a set of vertices S ⊂ V (G), the graph G\S is the graph induced by V (G)\S. The distance d(u, v) between two vertices u and v is the number of edges in the shortest path from u to v. We say that v is a d-neighbor of u if d(u, v) = d. Let Nd(v) be the set of d-neighbors of v. We will generally use the common term neighbor instead of 1-neighbor. A L(α1, α2, ..., αk)-labelling of a graphG is a function l : V (G)→ [0, λ] such that for any pair of vertices u and v if d(u, v) = d ≤ k then |l(u)− l(v)| ≥ αd. The problem is to find an L(α1, α2, ..., αk)-labelling of G that minimizes λ. We denote λα1,α2,...,αk(G) such minimal λ. For a sequence of non-negative integers S = (α1, α2, ..., αk), we will use the notation λS(G) instead of λα1,α2,...,αk(G). This problem arises from the channel assignement problem. The channel assignement problem is to assign a channel to each radio transmitter so that close transmitters do not interfer and such that we use the minimum span of frequency. Roberts proposed to assign channels such that “close” transmitters receive different channels and “very close” transmitters receive channels that are at least two channels apart. This is a L(2,1)-labelling of a graph G where the vertices are the transmitters, the “very close” transmitters are adjacent vertices and the “close” transmitters are vertices at distance two in G. Since the constraints between transmitters disminish with the distance, the L(α1, α2, ..., αk)-labelling of graph is interesting for this problem when the sequence α1, α2, ..., αk is decreasing. Many work has been done on L(2,1)-labeling since the first paper of J.R.Griggs and R.K.Yeh [7]. Many papers deal with bounding λα1,α2 for some family of graphs or given some graphs invariants such as χ(G) and ∆ (See for example [1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14]). In their paper [7], Griggs and Yeh proved that λ2,1(G) ≤ ∆+2∆ and made the following conjecture. Conjecture 1 For any graph G with maximal degree ∆ ≥ 2, λ2,1(G) ≤ ∆. Actually they proved it for ∆ = 2 and for graphs of diameter at most two. They also proved that determining λ2,1(G) is NP-complete. In this paper we focus on bounding λp,1 according to ∆. In [3] the authors gave an algorithm for the L(2,1)-labeling and improved the upper bound of λ2,1 to ∆ + ∆. In [4], with the same algorithm they obtained that λp,1(G) ≤ ∆ + (p − 1)∆. Let σ(S,∆) be the function