Vertex Coloring

This paper surveys the most important algorithmic and computational results on the Vertex Coloring Problem (VCP) and its generalizations. The first part of the paper introduces the classical models for the VCP, and discusses how these models can be used and possibly strengthened to derive exact and heuristic algorithms for the problem. Computational results on the best performing algorithms proposed in the literature are reported. The second part of the paper is devoted to some generalizations of the problem, which are obtained by considering additional constraints [Bandwidth (Multi) Coloring Problem, Bounded Vertex Coloring Problem] or an objective function with a special structure (Weighted Vertex Coloring Problem). The extension of the models for the classical VCP to the considered problems and the best performing algorithms from the literature, as well as the corresponding computational results, are reported.

In this paper, we deal with the notion of star coloring of graphs. A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored. We give the exact value of the star chromatic number of different families of graphs such as trees, cycles, complete bipartite graphs, outerplanar graphs and 2-dimensional grids. We also study and give bounds for the star chromatic number of other families of graphs, such as hypercubes, tori, d-dimensional grids, graphs with bounded treewidth and planar graphs.

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v, u, x, y) of vertices such that both (v, u, x) and (u, x, y) are paths of length two. The 3-arc graph of a graph G is defined to have the arcs of G as vertices such that two arcs uv, xy are adjacent if and only if (v, u, x, y) is a 3-arc of G. In this paper, we study the independence, domination and chromatic numbers of 3-arc graphs and obtain sharp lower and upper bounds for them. We introduce a new notion of arc-coloring of a graph in studying vertex-colorings of 3-arc graphs.

This paper surveys frequency assignment problems coming up in planning wireless communication services. It particularly focuses on cellular mobile phone systems such as GSM, a technology that revo­ lutionizes communication. Traditional vertex coloring provides a con­ ceptual framework for the mathematical modeling of many frequency planning problems. This basic form, however, needs various exten­ sions to cover technical and organizational

A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored. The star chromatic number of an undirected graph G, denoted by χs(G), is the smallest integer k for which G admits a star coloring with k colors. In this paper, we give the exact value of the star chromatic number of different families of graphs such as trees, cycles, complete bipartite graphs, outerplanar graphs and 2-dimensional grids. We also study and give bounds for the star chromatic number of other families of graphs, such as planar graphs, hypercubes, d-dimensional grids (d ≥ 3), d-dimensional tori (d ≥ 2), graphs with bounded treewidth and cubic graphs. We end this study by two asymptotic results, where we prove that, when d tends to infinity, (i) there exist graphs G of maximum degree d such that χs(G) = Ω( d 3 2 (log d) 1 2 ) and (ii) for any graph G of maximum degree d, χs(G) = O(d 3 2 ).

Given an undirected graph G = (V , E), the Vertex Coloring Problem (VCP) requires to assign a color to each vertex in such a way that colors on adjacent vertices are different and the number of colors used is minimized. In this paper, we present an exact algorithm for the solution of VCP based on the well-known Set Covering formulation of the problem. We propose a Branch-and-Price algorithm embedding an effective heuristic from the literature and some methods for the solution of the slave problem, as well as two alternative branching schemes. Computational experiments on instances from the literature show the effectiveness of the algorithm, which is able to solve, for the first time to proven optimality, five of the benchmark instances in the literature, and reduce the optimality gap of many others.

The local chromatic number of a graph was introduced in . It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs.

We give nontrivial bounds for the inductiveness or degeneracy of power graphs G k of a planar graph G. This implies bounds for the chromatic number as well, since the inductiveness naturally relates to a greedy algorithm for vertex-coloring the given graph. The inductiveness moreover yields bounds for the choosability of the graph. We show that the inductiveness of a square of a planar graph G is at most 9∆/5 , for the maximum degree ∆ sufficiently large, and that it is sharp. In general, we show for a fixed integer k ≥ 1 the inductiveness, the chromatic number, and the choosability of G k to be O(∆ k/2 ), which is tight.

In the paper we apply graph vertex coloring for verification of secret shares. We start from showing how to convert any graph into the number and vice versa. Next, theoretical result concerning properties of n-colorable graphs is stated and proven. From this result we derive graph coloring check-digit scheme. Feasibility of proposed scheme increases with the size of the number, which digits are checked and overall probability of errors. The check-digit scheme is used to build shares verification method that does not require cooperation of the third party. It allows implementing verification structure different from the access structure. It does not depend on particular secret sharing method. It can be used as long as the secret shares can be represented by numbers or graphs.

This paper analyzes some graph issues by using the symbolic program Mathematica and its version for the Web, webMathematica. In particular, we consider the problem of graph coloring: the assignment of colors to the vertices/edges of the graph such that adjacent vertices/edges are colored differently. In addition, we address the problem of obtaining the tenacity of binomial trees with Mathematica. Finally, we describe briefly an example of the application of our software to a scheduling problem.

We study the problem of finding occurrences of motifs in vertex-colored graphs, where a motif is a multiset of colors, and an occurrence of a motif is a subset of connected vertices whose multiset of colors equals the motif. This problem is a natural graph-theoretic pattern matching variant where we are not interested in the actual structure of the occurrence of the pattern, we only require it to preserve the very basic topological requirement of connectedness. We give two positive results and three negative results that together give an extensive picture of tractable and intractable instances of the problem.

In this paper, we propose a novel simple heuristic algorithm for scheduling the secondary link activation and provide a dynamic spectrum sharing in cognitive radio networks. This algorithm is presented for spectrum underlay where primary and secondary users transmit simultaneously on the same frequency bands in cognitive radio networks. The proposed algorithm is based on a graph-theoretical model. First, the

8 1 0 a r t i c l e i n f o 11 12 Keywords: 13 Vertex coloring problem 14 Learning automata 15 Cellular learning automata 16 1 7 a b s t r a c t 18 Vertex coloring problem is a combinatorial optimization problem in which a color is assigned to each ver-19 tex of the graph such that no two adjacent vertices have the same color. Cellular learning automata (CLA) 20 is an effective probabilistic learning model combining cellular automata and learning automata. Irregular 21 cellular learning automata (ICLA) is a generalization of cellular learning automata in which the restriction 22 of rectangular grid structure in traditional CLA is removed. In this paper, an ICLA-based algorithm is pro-23 posed for finding a near optimal solution of the vertex coloring problem. The proposed coloring algorithm 24 is a fully distributed algorithm in which each vertex chooses its optimal color based solely on the colors 25 selected by its adjacent vertices. The time complexity of the proposed algorithm is computed for finding a 26 1 1À optimal solution of the vertex coloring problem in an arbitrary graph. To show the superiority of our 27 proposed algorithm over the existing methods, simulation experiments have been conducted. The 28 obtained results show that the proposed algorithm outperforms the others in terms of the required num-29 ber of colors and running time of algorithm.

Extensions and variations of the basic problem of graph coloring are introduced. The problem consists essentially in finding in a graph G a k-coloring, i.e., a partition V 1 ,. .. , V k of the vertex set of G such that, for some specified neighborhoodÑ(v) of each vertex v, the number of vertices inÑ(v) ∩ V i is (at most) a given integer h i v. The complexity of some variations is discussed according toÑ(v), which may be the usual neighbors, or the vertices at distance at most 2, or the closed neighborhood of v (v and its neighbors). Polynomially solvable cases are exhibited (in particular when G is a special tree).

Many classes of graphs where the vertex coloring problem is polynomially solvable are known, the most prominent being the class of perfect graphs. However, the listcoloring problem is NP-complete for many subclasses of perfect graphs. In this work we explore the complexity boundary between vertex coloring and list-coloring on such subclasses of perfect graphs where the former admits polynomial-time algorithms but the latter is NP-complete. Our goal is to analyze the computational complexity of coloring problems lying "between" (from a computational complexity viewpoint) these two problems: precoloring extension, μ-coloring, and (γ , μ)-coloring.

In this paper we describe a new probabilistic approach to the role engineering process for RBAC. In particular, we address the issue of minimizing the number of roles, problem known in literature as the Basic Role Mining Problem (basicRMP). We leverage the equivalence of the above issue with the vertex coloring problem. Our main result is the proof that the minimum number of roles is sharply concentrated around its expected value. A further contribution is to show how this result can be applied as a stop condition when striving to find out an approximation for the basicRMP. We also show that the proposal can be used to decide whether it is advisable to undertake the efforts to renew an RBAC state. Note that both these applications can result in a substantial saving of resources. A thorough analysis using advanced probabilistic tools supports our results. Finally, further relevant research directions are also highlighted. 1

In this note, the authors generalize the ideas presented by Tucker in his proof of the Strong Perfect Graph Conjecture for (K4 − e)-free graphs in order to ÿnd a vertex v in G whose special neighborhood allows to extend a !(G)-vertex coloring of G − v to a !(G)-vertex coloring of G.

The representatives formulation for the vertex coloring problem is revisited to remove symmetry and new versions of facets derived from substructures of the graph are presented. In addition, a new class of facets is derived from independent sets of the graph. Finally, a comparison with the independent sets formulation is provided.

In a given graph G = (V , E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G if there exists a unique extension of the colors of S to a c ≥ χ (G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number. In this note, we study the chromatic number, the defining number and the strong defining number in some of the Harary graphs.

Given a graph G and a positive integer d, an L(d; 1)-labeling of G is a function f that assigns to each vertex of G a non-negative integer such that if two vertices u and v are adjacent, then |f(u) − f(v)|¿d; if u and v are not adjacent but there is a two-edge path between them, then

Study of algorithms and its design can be progressed in various dimensions. In this paper, we have a definite refinement of lower bound on the number of tracks required to route a channel. The attack is from a complementary viewpoint. Our algorithm succeeds to avoid all kinds of approximation. The approach performs exact mapping of the problem into graphical presentation and analyzes the graph taking help of mimetic algorithm, which uses combination of sequential and GA based vertex coloring. Performance of the algorithm depends on how effectively mimetic approach can be applied selecting appropriate values for the parameters to evaluate the graphical presentation of the problem. This viewpoint has immense contribution against sticking at local minima for this optimization problem. The finer result clearly exemplifies instances, which give better or at least the same lower bound in VLSI channel routing problem.

In this work we study a particular way of dealing with interference in combinatorial optimization models representing wireless communication networks. In a typical wireless network, co-channel interference occurs whenever two overlapping antennas use the same frequency channel, and a less critical interference is generated whenever two overlapping antennas use adjacent channels. This motivates the formulation of the minimum-adjacency vertex coloring problem which, given an interference graph G representing the potential interference between the antennas and a set of prespecified colors/channels, asks for a vertex coloring of G minimizing the number of edges receiving adjacent colors. We propose an integer programming model for this problem and present three families of facet-inducing valid inequalities. Based on these results, we implement a branch-and-cut algorithm for this problem, and we provide promising computational results.

We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is s-chromatic if it is colorable in s colors and any coloring of it uses at least s colors. The forcing chromatic number F χ (G) of an s-chromatic graph G is the smallest number of vertices which must be colored so that, with the restriction that s colors are used, every remaining vertex has its color determined uniquely. We estimate the computational complexity of F χ (G) relating it to the complexity class US introduced by Blass and Gurevich. We prove that recognizing if F χ (G) ≤ 2 is US-hard with respect to polynomial-time many-one reductions. Moreover, this problem is coNP-hard even under the promises that F χ (G) ≤ 3 and G is 3-chromatic. On the other hand, recognizing if F χ (G) ≤ k, for each constant k, is reducible to a problem in US via a disjunctive truth-table reduction.

In this paper, we address the decision problem for a sys- tem of monadic second-order logic interpreted over an!- layered temporal structure devoid of both a finest layer and a coarsest one (we call such a structure totally unbounded). We propose an automaton-theoretic method that solves the problem in two steps: first, we reduce the considered prob- lem to the

The purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both labelled and unlabelled m-ary cacti, according to i) the number of polygons,

We consider a multicast configuration with two sources, and translate the network code design problem to vertex coloring of an appropriately defined graph. This observation enables to derive code design algorithms and alphabet size bounds, as well as establish a connection with a number of well-known results from discrete mathematics that increase our insight in the different trade-offs possible for network coding.

We study connectivity, Hamilton path and Hamilton cycle decomposition, 4-edge and 3-vertex coloring for geometric graphs arising from pseudoline (a ne or projective) and pseudocircle (spherical) arrangements. While arrangements as geometric objects are well studied in discrete and computational geometry, their graph theoretical properties seem to have received little attention so far. In this paper we s h o w that they provide well structured examples of families of planar and projective-planar graphs with very interesting properties. Most prominently, spherical arrangements admit decompositions into two Hamilton cycles and 4-edge colorings, but other classes have i n teresting properties as well: 4-connectivity, 3-vertex coloring or Hamilton paths and cycles. We show a number of negative results as well: there are projective arrangements which cannot be 3-vertex colored. A n umb e r o f c o njectures and open questions accompany our results.

An extension of the basic image reconstruction problem in discrete tomography is considered: given a graph G = (V , E) and a family of chains P i together with vectors h(P i) = (h i 1 , ..., h i k), one wants to find a partition V 1 , ..., V k of V such that for each P i and each color j, |V j ∩ P i |= h i j. An interpretation in terms of scheduling is presented. We consider special cases of graphs and identify polynomially solvable cases; general complexity results are established in this case and also in the case where V 1 , ..., V k is required to be a proper ver-tex k-coloring of G. Finally, we examine also the case of (proper) edge k-colorings and determine its complexity status.

We study optimization problems that are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. Specifically, we focus on MAXIMUM INDEPENDENT SET, VERTEX COLORING, SET COVER, and BANDWIDTH. In recent years, many researchers design exact exponential-time algorithms for these and other hard problems. The goal is getting the time complexity still of order O(c n), but with the constant c as small as possible. In this work we extend this line of research and we investigate whether the constant c can be made even smaller when one allows constant factor approximation. In fact, we describe a kind of approximation schemes-trade-offs between approximation factor and the time complexity. We study two natural approaches. The first approach consists of designing a backtracking algorithm with a small search tree. We present one result of that kind: a (4r − 1)-approximation of BANDWIDTH in time O * (2 n/r), for any positive integer r. The second approach uses general transformations from exponential-time exact algorithms to approximations that are faster but still exponential-time. For example, we show that for any reduction rate r, one can transform any O * (c n)-time 1 algorithm for SET COVER into a (1 + ln r)approximation algorithm running in time O * (c n/r). We believe that results of that kind extend the applicability of exact algorithms for NP-hard problems. Classification: Algorithms and data structures; "fast" exponential-time algorithms

Communications problems that involve frequency interference, such as the channel assignment problem in the design of cellular telephone networks, can be cast as graph coloring problems in which the frequencies (colors) assigned to an edge's vertices interfere if they are too similar. The paper considers situations modeled by vertex-coloring d-regular graphs with n vertices using a color set f1; 2; : : :; ng, where colors i and j are said to interfere if their circular distance minfji ? jj; n ? ji ? jjg does not exceed a given threshold value. Given a d-regular graph G and threshold , an interference-minimizing coloring is a coloring of vertices that minimizes the number of edges that interfere. Let I (G) denote the minimum number of interfering edges in such a coloring of G. For most triples (n; ; d) we determine the minimum value of I (G) over all d-regular graphs, and nd graphs that attain it. In determining when this minimum value is 0, we prove that for r 3 there exists a d-regular graph G on n vertices that is r-colorable whenever d (1 ? 1 r)n ? 1 and nd is even. We also study the maximum value of I (G) over all d-regular graphs, and nd graphs that attain this maximum in many cases.