Wireless sensor networks benefit from communication protocols that reduce power requirements by avoiding frame collision. Time Division Media Access methods schedule transmission in slots to avoid collision, however these methods often lack scalability when implemented in ad hoc networks subject to node failures and dynamic topology. This paper reports a distributed algorithm for TDMA slot assignment that is self-stabilizing to transient faults and dynamic topology change. The expected local convergence time is O(1) for any size network satisfying a constant bound on the size of a node neighborhood.

For any non negative real values h and k, an L(h, k)-labeling of a graph G = (V, E) is a function L: V → IR such that |L(u) − L(v) | ≥ h if (u, v) ∈ E and |L(u) − L(v) | ≥ k if there exists w ∈ V such that (u, w) ∈ E and (w, v) ∈ E. The span of an L(h, k)-labeling is the difference between the largest and the smallest value of L, so it is not restrictive to assume 0 as the smallest value of L. We denote by λh,k(G) the smallest real λ such that graph G has an L(h, k)-labeling of span λ. The aim of the L(h, k)-problem is to satisfy the distance constraints using the minimum span. In this paper, we study L(h, k)-labeling problem on regular grids of degree 3, 4, 6, and 8 solving several open problems left in the literature. Keywords: L(h,k)-labeling, triangular grids, hexagonal grids, squared grids, octagonal grids. 1

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The L(h, k)-labeling is an assignment of non negative integer labels to the nodes of a graph such that ’close ’ nodes have labels which differ by at least k, and ’very close ’ nodes have labels which differ by at least h. The span of an L(h, k)-labeling is the difference between the largest and the smallest assigned label. We study L(h, k)-labelings of cellular, squared and hexagonal grids, seeking those with minimum span for each value of k and h ≥ k. The L(h, k)labeling problem has been intensively studied in some special cases, i.e. when k = 0 (vertex coloring), h = k (vertex coloring the square of the graph) and h = 2k (radio- or λ-coloring) but no results are known in the general case for regular grids. In this paper, we completely solve the L(h, k)-labeling problem on cellular grids, finding exact values of the span for each value of h and k; only in a small interval we provide different upper and lower bounds. For the sake of completeness, we study also hexagonal and squared grids. Keywords: L(h, k)-labeling, cellular grids, triangular grids, hexagonal grids, squared grids. 1

In this paper, we address the problem of λ labelings, that was introduced in the context of frequency assignment for telecommunication networks. In this model, stations within a given radius r must use frequencies that differ at least by a value p, while stations that are within a larger radius r ′> r must use frequencies that differ by at least another value q. The aim is to minimize the span of frequencies used in the network. This can be modeled by a graph coloring problem, called the L(p, q) labeling, where one wants to label vertices of the graph G modeling the network by integers in the range [0; M], in such a way that (1) neighbors in G are assigned colors differing by at least p and (2) vertices at distance 2 in G are assigned colors differing by at least q, while minimizing the value of M. M is then called the λ number of G, and is denoted by λ p q(G). In this paper, we study the L(p, q) labeling for a specific class of networks, namely the d-dimensional grid Gd = G[n1, n2... nd]. We give bounds on the value of the λ number of an L(p, q) labeling for any d ≥ 1 and p, q ≥ 0. Some of these results are optimal (namely, in the

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For positive integers j ≥ k, an L(j,k)-labeling of a digraph D is a function f from V (D) into the set of nonnegative integers such that |f(x) − f(y) | ≥ j if x is adjacent to y in D and |f(x) − f(y) | ≥ k if x is of distant two to y in D. Elements of the image of f are called labels. The L(j,k)-labeling problem is to determine the � λj,knumber � λj,k(D) of a digraph D, which is the minimum of the maximum label used in an L(j,k)-labeling of D. This paper studies � λj,k-numbers of digraphs. In particular, we determine � λj,k-numbers of digraphs whose longest dipath is of length at most 2, and � λj,k-numbers of ditrees having dipaths of length 4. We also give bounds for � λj,knumbers of bipartite digraphs whose longest dipath is of length 3. Finally, we present a linear-time algorithm for determining � λj,1-numbers of ditrees whose longest dipath is of length 3. Keywords. L(j,k)-labeling, digraph, ditree, homomorphism, algorithm.

tree and path backbones

Abstract—This paper describes a set of TDMA MAC protocols for wireless sensor networks that can achieve near-optimal throughput and good latency for regular periodic data delivery. The protocol is based on a novel graph coloring technique called the Color Constraint Heuristic (CCH) The paper describes a centralized TDMA slot assignment algorithm, Centralized Slot Assignment (CSA-CCH), that uses CCH. It then describes a

We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph G = (V, E) and a spanning subgraph H of G (the backbone of G), a λ-backbone coloring for G and H is a proper vertex coloring V → {1, 2,...} of G in which the colors assigned to adjacent vertices in H differ by at least λ. The algorithmic and combinatorial properties of backbone colorings have been studied for various types of backbones in a number of papers. The main outcome of earlier studies is that the minimum number ℓ of colors for which such colorings V → {1, 2,..., ℓ} exist in the worst case is a factor times the chromatic number (for all studied types of backbones). We show here that for split graphs and matching or star backbones, ℓ is at most a small additive constant (depending on λ) higher than the chromatic number. Our proofs combine algorithmic and combinatorial arguments. We also indicate other graph classes for which our results imply better upper bounds on ℓ than the previously known bounds.