Given a non-decreasing sequence S = (s 1 , s 2 , . . . , s k ) of positive integers, an Spacking edge-coloring of a graph G is a partition of the edge set of G into k subsets {X 1 , X 2 , . . . , X k } such that for each 1 ≤ i ≤ k, the distance between two distinct edges e, e ′ ∈ X i is at least s i + 1. This paper studies S-packing edgecolorings of cubic graphs. Among other results, we prove that cubic graphs having a 2-factor are (1, 1, 1, 3, 3)-packing edge-colorable, (1, 1, 1, 4, 4, 4, 4, 4)-packing edgecolorable and (1, 1, 2, 2, 2, 2, 2)-packing edge-colorable. We determine sharper results for cubic graphs of bounded oddness and 3-edge-colorable cubic graphs and we propose many open problems.
The context of this paper is programmable matter, which consists of a set of computational elements, called particles, in an infinite graph. The considered infinite graphs are the square, triangular and king grids. Each particle occupies one vertex, can communicate with the adjacent particles, has the same clockwise direction and knows the local positions of neighborhood particles. Under these assumptions, we describe a new leader election algorithm affecting a variable to the particles, called the k-local identifier, in such a way that particles at close distance have each a different k-local identifier. For all the presented algorithms, the particles only need a O(1)-memory space.there exists a probabilistic algorithm that determine a leader (and in particular for a ring) with probability 1 [5].Several projects aim to build programmable matter prototypes. One of such projects [20,23], financed by the french National Agency for Research, aims to build cuboctahedral particles able to deform them-selves in order to move. This project can be split in two phases, one consists in manufacturing the hardware of prototype matters, the second consists in proposing algorithms for programmable matter. The final goal of this project is to sculpt a shape-memory polymer sheet with programmable matter. In the continuity of the algorithm phase of this project [20], we propose algorithms for the self-configuration, i.e., in order to create identifiers and spanning trees.In the context of programmable matter [3,4,14,18,23,24], it is supposed that a network can contain several millions of modules and that each module has possibly a nano-centimeter size. These two facts lead us to believe that even a O(log(n))-space memory for each module, n being the number of modules, is not technically possible. Also, because of the large number of modules, it can be very challenging and time consuming to implement a unique identity to the modules when they are created. In this context, we suppose that the modules can not store a unique identity, i.e., that the network is anonymous. In this paper we propose deterministic O(1)-space memory algorithms to determine a leader in the network and to create k-local identifiers of the particles. A k-local identifier is a variable affected to each module of the network which is different for every two modules at distance at most k. Note that leader election [5, 13] plays a significant role in numerous problems of programmable matter.Our contribution is the following: we introduce a leader election algorithm based on local computations and simple to implement. This algorithm works when the structure the particles form has no hole (see Section 3). Also, since the algorithm can be described as a sequence of local computations, its limits (message complexity, required memory-space, etc) are easy to analyze. We present a distributed algorithm to construct a spanning tree in the context of programmable matter and, also, a distributed algorithm to re-organize the port numbers of the particles. Finally, we presen...
Although it has recently been proved that the packing chromatic number is unbounded on the class of subcubic graphs, there exists subclasses in which the packing chromatic number is finite (and small). These subclasses include subcubic trees, base-3 Sierpiski graphs and hexagonal lattices. In this paper we are interested in the packing chromatic number of subcubic outerplanar graphs. We provide asymptotic bounds depending on structural properties of the outerplanar graphs and determine sharper bounds for some classes of subcubic outerplanar graphs.
This work establishes the complexity class of several instances of the S-packing coloring problem: for a graph G, a positive integer k and a nondecreasing list of integers S = (s1, . . . , s k ), G is S-colorable if its vertices can be partitioned into sets Si, i = 1, . . . , k, where each Si is an si-packing (a set of vertices at pairwise distance greater than si). In particular we prove a dichotomy between NP-complete problems and polynomial-time solvable problems for lists of at most four integers.Let S k d be a list only containing k integers d. The problem S k 1 -COL corresponds to the k-coloring problem which is known to be NP-complete for k ≥ 3. The S-coloring generalizes coloring with distance constraints like the packing coloring or the distance coloring of a graph. We denote by P-COL, the problem (1, 2, . . . , k)-COL for a graph G and an integer k (with G and k as input). The packing chromatic number [9] of G is the least integer k such that G is (1, 2, . . . , k)-colorable. A series of works [4,6,8,9] considered the packing chromatic number of infinite grids. The d-distance chromatic number [13] of G is the least integer k such that G is S k d -colorable. Initially, the concept of S-coloring has been introduced by Goddard et al. [9] and Fiala et al. [7]. The S-coloring problem was considered in other papers [10,11].The S-coloring problem, with |S| = 3 has been introduced by Goddard et al. [9] in order to determine the complexity of the packing chromatic number when k = 4. Moreover, Goddard and Xu [10] have proven that for |S| = 3, S-COL is NP-complete if s 1 = s 2 = 1 or if s 1 = 1 and s 2 = s 3 = 2 and polynomial-time solvable otherwise. About the complexity of S-COL, Fiala et al. [7] have proven that P-COL is NP-complete for trees and Argiroffo et al. [1,2] have proven that P-COL is polynomial-time solvable on some classes of graphs.In the second section, for a list S of three integers, we determine the family of S-colorable trees. Moreover, we determine dichotomies on cubic graphs, subcubic graphs and bipartite graphs. In the third section, we determine polynomialtime solvable and NP-complete instances of S-COL, for unfixed size of lists. We use these results to determine a dichotomy between NP-complete instances and polynomial-time solvable instances of S-COL for |S| ≤ 4.Note that for any nondecreasing list of integers S, we have S-COL in NP.
An i-packing in a graph G is a set of vertices at pairwise distance greater than i. For a nondecreasing sequence of integers S = (s1, s2, . . .), the S-packing chromatic number of a graph G is the least integer k such that there exists a coloring of G into k colors where each set of vertices colored i, i = 1, . . . , k, is an si-packing. This paper describes various subdivisions of an i-packing into j-packings (j > i) for the hexagonal, square and triangular lattices. These results allow us to bound the Spacking chromatic number for these graphs, with more precise bounds and exact values for sequences S = (si, i ∈ N * ), si = d + ⌊(i − 1)/n⌋.
Given a graph G and a nondecreasing sequence S = (s1, . . . , s k ) of positive integers, the mapping c : V (G) −→ {1, . . . , k} is called an Spacking coloring of G if for any two distinct vertices x and y in c −1 (i), the distance between x and y is greater than si. The smallest integer k such that there exists a (1, 2, . . . , k)-packing coloring of a graph G is called the packing chromatic number of G, denoted χρ(G). The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs.In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by 7. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a (1, 2, 2, 2)-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a (1, 2, 2, 2)-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a (1, 2, 2, 3)-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an Spacking coloring for S = (1, 3, . . . , 3), where 3 appears ∆ times (∆ being the maximum degree of vertices), and if one of the integers 3 is replaced by 4 in the sequence S. Also, there exist outerplanar bipartite graphs that do not admit an S-packing coloring.
The face-centered cubic grid is a three dimensional 12-regular infinite grid. This graph represents an optimal way to pack spheres in the three-dimensional space. In this grid, the vertices represent the spheres and the edges represent the contact between spheres. We give lower and upper bounds on the chromatic number of the d th power of the face-centered cubic grid. In particular, in the case d = 2 we prove that the chromatic number of this grid is 13. We also determine sharper bounds for d = 3 and for subgraphs of of the face-centered cubic grid.
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