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.

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 ).

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.

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.

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.

A special DNA computer was designed to solve the vertex coloring problem. The main body of this kind of DNA computer was polyacrylamide gel electrophoresis which could be classified into three parts: melting region, unsatisfied solution region and solution region. This polyacrylamide gel was connected with a controllable temperature device, and the relevant temperature was T m1, T m2 and T m3, respectively. Furthermore, with emphasis on the encoding way, we succeeded in performing the experiment of a graph with 5 vertices. In this paper we introduce the basic structure, the principle and the method of forming the library DNA sequences.

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.

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.

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.

In this paper we study a new notion of coloring type of graph, namely a local
irregularity vertex coloring. We define is called vertex
irregular -labeling and where . By
a local irregularity vertex coloring, we define a condition for if for every
and vertex irregular labeling .
The chromatic number of local irregularity vertex coloring of , denoted by ,
is the minimum cardinality of the largest label over all such local irregularity vertex

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.

We consider vertex coloring of an acyclic digraph G in such a way that two vertices which have a common ancestor in G receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding down-chromatic number and derive an upper bound as a function of D( G), the maximum number of descendants of a given vertex, and the degeneracy of the corresponding hypergraph. Finally we determine an asymptotically tight upper bound of the down-chromatic number in terms of the number of vertices of G and D( G).

In this paper uniquely list colorable graphs are studied. A graph G is called to be uniquely k-list colorable if it admits a k-list assignment from which G has a unique list coloring. The minimum k for which G is not uniquely k-list colorable is called the m-number of G. We show that every triangle-free uniquely colorable graph with chromatic number k + 1, is uniquely k-list colorable. A bound for the m-number of graphs is given, and using this bound it is shown that every planar graph has m-number at most 4. Also we introduce list criticality in graphs and characterize all 3-list critical graphs. It is conjectured that every χ ′ ℓ -critical graph is χ ′ -critical and the equivalence of this conjecture to the well known list coloring conjecture is shown.

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.

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.

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.

A cyclic coloring is a vertex coloring such that vertices in a face receive di erent colors. Let be the maximum face degree of a graph. This article shows that plane graphs have cyclic 9 5 -colorings, improving results of Ore and Plummer, and of Borodin. The result is mainly a corollary of a best-possible upper bound on the minimum cyclic degree of a vertex of a plane graph in terms of its maximum face degree. The proof also yields results on the projective plane, as well as for d-diagonal colorings. Also, it is shown that plane graphs with = 5 have cyclic 8-colorings. This result and also the 9 5 result are not necessarily best possible.

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.

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 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.

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

Vertex-colorings, edge-colorings and total-colorings of the Sierpiński gasket graphs S n , the Sierpiński graphs S(n, k), graphs S + (n, k), and graphs S ++ (n, k) are considered. In particular, χ ′′ (S n ), χ ′ (S(n, k)), χ(S + (n, k)), χ(S ++ (n, k)), χ ′ (S + (n, k)), and χ ′ (S ++ (n, k)) are determined.

The vertex coloring problem has been the subject of extensive research for many years. Driven by application potential as well as computational challenge, a variety of methods have been proposed for this difficult class of problems. Recent successes in the use of the unconstrained quadratic programming (UQP) model as a unified framework for modeling and solving combinatorial optimization problems have motivated a new approach to the vertex coloring problem. In this paper we present a UQP approach to this problem and illustrate its attractiveness with preliminary computational experience.

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 classi cation 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, ii) the vertex-color distribution, iii) the vertex-degree distribution of each color. We also enumerate m-ary cacti according to the order of their automorphism group. Using a generalization of Otter's formula, we express the species of m-ary cacti in terms of rooted and of pointed cacti. A variant of the m-dimensional Lagrange inversion is then used to enumerate these structures. The method of Liskovets for the enumeration of unrooted planar maps can also be adapted to m-ary cacti.

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.

The unique structural configuration found in human foot allows easy walking. Similar movement is hard to imitate even for an ape. It is obvious that human ambulation relates to the foot structure itself. Suppose the bones are represented as vertices and the joints as edges. This leads to the development of a special graph that represents human foot. On a footprint there are point-ofcontacts which have contact with the ground. It involves specific vertices. Theoretically, for an ideal ambulation, these points provide reactions onto the ground or the static equilibrium forces. They are arranged in sequence in form of a path. The ambulating footprint follows this path. Having the human foot graph and the path crossbred, it results in a representation that describes the profile of an ideal ambulation. This profile cites the locations where the point-of-contact experience normal reaction forces. It highlights the significant of these points.

http://cab.um.edu.my/images/cab/doc/Publications/Journal2009/banihashim.pdf

We study the off and on-line versions of the well known problem of scheduling a set of n independent multiprocessor tasks with prespecified processor allocations on a set of identical processors in order to minimize the makespan. Recently, in [12], it has been proven that in the case when all tasks have unit processing time the problem cannot be approximated within a factor of m 1– ∈, neither for some ∈ > 0, unless P= NP; nor for any ∈ > 0, unless NP=ZPP. For this special case we give a simple algorithm based on the classical first-fit technique. We analyze the algorithm for both tasks arrive over time and tasks arrive over list on-line scheduling versions, and show that its competitive ratio is bounded by 2√m and 2√m + 1, respectively. Here we also use some preliminary results on (vertex) coloring of k-tuple graphs. For the case of arbitrary processing times, we show that any algorithm which uses the first-fit technique cannot be better than m competitive. Then, by using our split-round technique, we give a 3√m-approximation algorithm for the off-line version of the problem. Finally, by using some ideas from [20], we adapt the algorithm to the on-line case, in the paradigm of tasks arriving over time in which the existence of a task is unknown until its release date, and show that its competitive ratio is bounded by 6√m. Due to the conducted experimental results, we conclude that our algorithms can perform well in practice.

Timetabling problems are present in all types of schools. The research in this area is still very active; of the 19 selected contributions of PATAT 2004 ([1]), 12 are dedicated to Educational Timetabling. These problems can often be mod-eled by a graph coloring problem. Here ...

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.

A defining set (of vertex coloring) of a graph G is a set of vertices S with an assignment of colors to its elements which has a unique completion to a proper coloring of G. We define a minimal defining set to be a defining set which does not properly contain another defining set. If G is a uniquely vertex colorable graph, clearly its minimum defining sets are of size #(G) 1. It is shown that for a coloring of G, if all minimal defining sets of G are of size #(G) 1, then G is a uniquely vertex colorable graph.