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This study focuses on two heuristic algorithms for the graph vertex coloring problem: the sequential (greedy) coloring algorithm (SCA) and the Welsh–Powell algorithm (WPA). The code of the algorithms is presented and discussed. The methodology and conditions of the experiments are presented. The execution time of the algorithms was calculated as the average of four different starts of the algorithms for all analyzed graphs, taking into consideration the multitasking mode of the operating system. In the graphs with less than 600 vertices, in 90% of cases, both algorithms generated the same solutions. In only 10% of cases, the WPA algorithm generates better solutions. However, in the graphs with more than 1,000 vertices, in 35% of cases, the WPA algorithm generates better solutions. The results show that the difference in the execution time of the algorithms for all graphs is acceptable, but the quality of the solutions generated by the WPA algorithm in more than 20% of cases is better compared to the SC algorithm. The results also show that the quality of the solutions is not related to the number of iterations performed by the algorithms.

Graph coloring has a wide range of real world applications, such as in the operations research, communication network, computational biology and compiler optimization fields. In our recent work [1], we propose a divide-and-conquer approach for graph coloring, called VColor. Such an approach has three generic subroutines. (i) Graph partition subroutine: VColor partitions a graph G into a vertex cut partition (VP), which comprises a vertex cut component (VCC) and small non-overlapping connected components (CCs). (ii) Component coloring subroutine: VColor colors the VCC and the CCs by efficient algorithms. (iii) Color combination subroutine: VColor combines the local colors by exploiting the maximum matchings of color combination bigraphs (CCBs). VColor has revealed some major bottlenecks of efficiency in these subroutines. Therefore, in this paper, we propose VColor*, an approach which addresses these efficiency bottlenecks without using more colors both theoretically and experimentally. The technical novelties of this paper are the following. (i) We propose the augmented VP to index the crossing edges of the VCC and the CCs and propose an optimized CCB construction algorithm. (ii) For sparse CCs, we propose using a greedy coloring algorithm that is of polynomial time complexity in the worst case, while preserving the approximation ratio. (iii) We propose a distributed graph coloring algorithm. Our extensive experimental evaluation on real-world graphs confirms the efficiency of VColor*. In particular, VColor* is 20X and 50X faster than VColor and uses the same number of colors with VColor on the Pokec and PA datasets, respectively. VColor* also significantly outperforms the state-of-the-art graph coloring methods.

Graph coloring is one of the fundamentally known NP-complete problem. Several heuristics have been developed to solve this problem, among which greedy coloring is the most naturally used one. In greedy coloring, vertices are traversed following an order and hence performance of it highly depends on finding a good order. In this paper, we propose an incremental search heuristic (ISH) which considers some ρ1 random orders and for each of them it calls a selective search (SS) procedure with parameter ρ2. Given an order, SS considers only those orders which produces equal or less number of distinct colors than a given order. We showed that those orders can be partitioned into disjoint subsets of equivalent orders. To make effective search, SS selects and evaluates only one order from such a subset. Analytically we have shown that ISH can solve the graph coloring instances on sparse graph in expected polynomial time. Through simulations, we have evaluated ISH on 86 challenging benchmarks and compare results with state of the art existing algorithms. We observed that ISH significantly outperforms existing algorithms specially for sparse graphs and also produces reasonably good results for others.

A lid-coloring (locally identifying coloring) of a graph is a proper coloring such that, for any edge uv, if u and v have distinct closed neighborhoods, then the set of colors used on vertices of the closed neighborhoods of u and v are distinct. The lid-chromatic number is the minimum number of colors used in a lid-coloring. In this paper we prove a relation between lid-coloring and a variation, called strong lid-coloring. With this, we obtain linear time algorithms to calculate the lid-chromatic number for some classes of graphs with few P 4 's, such as cographs, P 4 -sparse graphs and ( q , q − 4 ) -graphs. We also prove that the lid-chromatic number is O ( n 1 − e ) -inapproximable in polynomial time for every e > 0 , unless P = NP .

In this paper, a novel GPS-assisted grouping scheme is proposed to reduce the operational cost of cognitive radio networks with femtocells. This scheme allows the cognitive base station (CBS) to determine the minimum number of channels to be rented from the primary user (PU) networks by utilizing greedy graph coloring and frequency re-use. The proposed scheme is optimized, in terms of the distances between the FBSs in the same group, and extended to the co-channel deployment case (i.e., when the macrocell secondary users (MSUs) and the FBSs are operating on the same spectrum band). Moreover, the performance of the scheme is analyzed in terms of the average number of CBS channels, average outage probability, and complexity. Furthermore, two benchmark schemes are devised and compared to the distance-based greedy coloring scheme; namely, the optimal distance-based and the profit maximization schemes. Simulation results show that the performance of the distance-based greedy coloring scheme, in terms of reducing the number of channels to be purchased from the PU networks, approaches that of the optimal scheme in high interference environments.

In this survey the following types of colorings of plane graphs are discussed, both in their vertex and edge versions: facially proper coloring, rainbow coloring, antirainbow coloring, loose coloring, polychromatic coloring, l -facial coloring, nonrepetitive coloring, odd coloring, unique-maximum coloring, WORM coloring, ranking coloring and packing coloring. In the last section of this paper we show that using the language of words these different types of colorings can be formulated in a more general unified setting.

Coupon coloring is a new coloring which has many applications. A k -coupon coloring of a graph G is a k -coloring of G by colors [ k ] = { 1 , 2 , … , k } such that the neighborhood of every vertex of G contains vertices of all colors from [ k ]. The maximum integer k for which a k -coupon coloring exists is called the coupon coloring number of G , and it is denoted by χ c ( G ). In this paper, we studied the coupon coloring of cographs, which are graphs that can be generated from the single vertex graph K 1 by complementation and disjoint union, and have applications in many interesting problems. We use the cotree representation of a cograph to give a polynomial time algorithm to color the vertices of a cograph, and then prove that this coloring is a coupon coloring with maximum colors, hence get the coupon coloring numbers of the cograph.

Feder and Subi conjectured that for any 2 -coloring of the edges of the n -dimensional cube, there is an antipodal pair of vertices connected by a path that changes color at most once. They proved that if the coloring is such that there are no properly edge colored 4 -cycles, the conjecture is true, without a color change. We generalize their theorem by weakening the assumption on the coloring. Our method can be applied to a similar question on any graph, if the condition on the coloring is satisfied. We solve the corresponding problem on the toroidal grid of size 2 a × 2 b .

We consider an extremal problem motivated by a article of Balogh [J. Balogh, A remark on the number of edge colorings of graphs, European Journal of Combinatorics 27, 2006, 565–573], who considered edge-colorings of graphs avoiding fixed subgraphs with a prescribed coloring. More precisely, given r≥t≥2, we look for n-vertex graphs that admit the maximum number of r-edge-colorings such that at most t−1 colors appear in edges incident with each vertex, that is, r-edge-colorings avoiding rainbow-colored stars with t edges. For large n, we show that, with the exception of the case t=2, the complete graph Kn is always the unique extremal graph. We also consider generalizations of this problem.

Objective: To improve the Edge Cover based Graph Coloring Algorithm (ECGCA) using independent set by incorporating parallel computing aspects in algorithm. Finding optimum time complexity is one of the main objectives of this paper. Methods/Statistical Analysis: This paper introduced some modification in ECGCA. Algorithm is implemented and tested using Java programming language. Java multithreading concept is used to achieve parallel computing in algorithm. DIMACS graph instances are used to test algorithm. Finding: Algorithm is tested on more than 75 DIMACS graph instances. To analyze the time complexity, execution time of algorithm in seconds is calculated by program. Algorithm is tested on different DIMACS graph instances. Test data is analyze in this paper and found that proposed algorithm executed in optimum time for large graphs. This paper also compared parallel algorithm and found that proposed parallel algorithm is less time complex than sequential algorithm. Most of the exact graph coloring algorithms are not suitable for large graph (more than 100 vertices) but proposed algorithm is tested on many large graphs and high execution success rate of algorithm is achieved. Application: This paper shows the experimental results of different type of application data. It means this algorithm can be used for maximum types of applications.

In this note we consider two coloring problems in mixed graphs, i.e., graphs containing edges and arcs, which arise from scheduling problems where disjunctive and precedence constraints have to be taken into account. We show that they are both NP-complete in cubic planar bipartite mixed graphs, which strengthens some results of Ries and de Werra (2008) [9].

A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different fromthe sumof colors taken on the edges incident to v. Let χ′Σ(G) denote the smallest value k in such a coloring of G. This parameter makes sense for graphs containing no isolated edges (we call such graphs normal). The maximum average degree mad(G) of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad(G) < 5/2, then χ′Σ(G) ≤ 5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.

Given an undirected simple graph gg, an integer kk, and a cost cijcij for each pair of non-adjacent vertices in gg, a robust coloring of gg is the assignment of kk colors to vertices such that adjacent vertices get different colors and the total penalty of the pairs of vertices having the same color is minimum. The problem has applications in fields such as timetabling and scheduling. We present a new formulation for the problem, which extends an existing formulation for the graph coloring problem. We also discuss a column generation based solution method. We report computational study on the performance of alternative formulations and the column generation method.

International audience; In Vertex Coloring Problems, one is required to assign a color to each vertex of an undirected graph in such a way that adjacent vertices receive different colors, and the objective is to minimize the cost of the used colors. In this work we solve four different coloring problems formulated as Maximum Weight Stable Set Problems on an associated graph. We exploit the transformation proposed by Cornaz and Jost (2008), where given a graph GG, an auxiliary graph View the MathML sourceGˆ is constructed, such that the family of all stable sets of View the MathML sourceGˆ is in one-to-one correspondence with the family of all feasible colorings of GG. The transformation in Cornaz and Jost (2008) was originally proposed for the classical Vertex Coloring and the Max-Coloring problems; we extend it to the Equitable Coloring Problem and the Bin Packing Problem with Conflicts. We discuss the relation between the Maximum Weight Stable formulation and a polynomial-size formulation for the VCP, proposed by Campêlo et al. (2008) and called the Representative formulation. We report extensive computational experiments on benchmark instances of the four problems, and compare the solution method with the state-of-the-art algorithms. By exploiting the proposed method, we largely outperform the state-of-the-art algorithm for the Max-coloring Problem, and we are able to solve, for the first time to proven optimality, 14 Max-coloring and 2 Equitable Coloring instances.

We demonstrate that for every positive integer $\Delta$, every K\_4-minor-free graph with maximum degree $\Delta$ admits an equitable coloring with k colors wherek $\ge$ ($\Delta$+3)/2. This bound is tight and confirms a conjecture by Zhang and Whu. We do not use the discharging method but rather exploit decomposition trees of K 4-minor-free graphs.

We show two results related to the Hamiltonicity and $k$-Path algorithms in undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10]. First, we demonstrate that the technique used can be generalized to finding some $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph in $O^*(1.657^k2^{l/2})$ time. It can be applied as a subroutine to solve the $k$-Internal Spanning Tree ($k$-IST) problem in $O^*(\min(3.455^k, 1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^*(2^n)$. Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for $k$-Path and Hamiltonicity in any graph of maximum degree $\Delta=4,\ldots,12$ or with vector chromatic number at most 8.

Nilpotent adjacency matrix methods have proven useful for counting self-avoiding walks (paths, trails, cycles, and circuits) in finite graphs. In the current work, these methods are extended for the first time to problems related to graph colorings. Nilpotent-algebraic formulations of graph coloring problems include necessary and sufficient conditions for k-colorability, enumeration (counting) of heterogeneous and homogeneous paths, trails, cycles, and circuits in colored graphs, and a matrix-based greedy coloring algorithm. Introduced here also are the orthozeons and their application to counting monochromatic self-avoiding walks in colored graphs. The algebraic formalism easily lends itself to symbolic computations, and Mathematica-computed examples are presented throughout.

Let G be a planar graph without 4-cycles and 5-cycles and with maximum degree Δi¾ź32. We prove that i¾źi¾źG2i¾źΔ+3. For arbitrarily large maximum degree Δ, there exist planar graphs GΔ of girth 6 with i¾źGΔ2=Δ+2. Thus, our bound is within 1 of being optimal. Further, our bound comes from coloring greedily in a good order, so the bound immediately extends to online list-coloring. In addition, we prove bounds for Lp,q-labeling. Specifically, λ2,1Gi¾źΔ+8 and, more generally, λp,qGi¾ź2q-1Δ+6p-2q-2, for positive integers p and q with pi¾źq. Again, these bounds come from a greedy coloring, so they immediately extend to the list-coloring and online list-coloring variants of this problem.

We consider list coloring of planar graphs without cycles of length in {4, …, 8}. List coloring is a generalization of the classical vertex coloring problem where each vertex has a list of colors associated with it. The goal is to proper vertex color the graph, such that each vertex gets a color available in its list. In this note, we prove that it is possible to 3-list color planar graphs without cycles of length in {4, …, 8} and with restrictions on 9-cycles.

We propose two vertex-coloring problems for graphs, endorsing the spectrum of colors with a matrix of interferences between pairs of colors. In the Threshold Spectrum Coloring problem, the number of colors is fixed and the aim is to minimize the maximum interference at a vertex (interference threshold). In the Chromatic Spectrum Coloring problem, a threshold is settled and the aim is to minimize the number of colors (among the available ones) for which respecting that threshold is possible. We prove general upper bounds for the solutions to each problem, valid for any graph and any matrix of interferences. We also show that both problems are NP-hard and perform experimental results, proposing a DSATUR-based heuristic for each problem, in order to study the gap between the theoretical upper bounds and the values obtained., Ministerio de Economía y Competitividad, Comunidad de Madrid

A total coloring of a graph is an assignment of colors to all the elements of the graph such that no two adjacent or incident elements receive the same color. In this paper, we prove the tight bound of Behzad and Vizing conjecture on total coloring for Compound graph of G and H, where G and H are any graphs.

Despite the fact that many vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not "under control" from a polyhedral point of view. The equivalence between optimization and separation suggests the existence of integer programming formulations for these problems whose associated polytopes admit elegant characterizations. In this work we address this issue. As a starting point, we focus our attention on the well-known standard formulation for the classical vertex coloring problem. We present some general results about this formulation and we show that the vertex coloring polytope associated to this formulation for a graph G and a set of colors C corresponds to a face of the stable set polytope of a particular graph S G C . We further study the perfectness of S G C showing that when ź C ź 2 , this graph is perfect if and only if G is a block graph, from which we deduce a complete characterization of the associated coloring polytopes for block graphs. We also derive a new family of valid inequalities generalizing several known families from the literature and we conjecture that this family is sufficient to completely describe the vertex coloring polytope associated to cacti graphs.

Given a graph G , its k -coloring graph is the graph whose vertex set is the proper k -colorings of the vertices of G with two k -colorings adjacent if they differ at exactly one vertex. In this paper, we consider the question: Which graphs can be coloring graphs? In other words, given a graph H , do there exist G and k such that H is the k -coloring graph of G ? We will answer this question for several classes of graphs and discuss important obstructions to being a coloring graph involving order, girth, and induced subgraphs.

Graph coloring is a classical NP-hard combinatorial optimization problem with many practical applications. A broad range of heuristic methods exist for tackling the graph coloring problem: from fast greedy algorithms to more time-consuming metaheuristics. Although the latter produce better results in terms of minimizing the number of colors, the former are widely employed due to their simplicity. These heuristic methods are centralized since they operate by using complete knowledge of the graph. However, in real-world environmets where each component only interacts with a limited number of other components, the only option is to apply decentralized methods. This paper explores a novel and simple algorithm for decentralized graph coloring that uses a fixed number of colors and iteratively reduces the edge conflicts in the graph. We experimentally demonstrate that, for most of the tested instances, the new algorithm outperforms a recent and very competitive algorithm for decentralized graph coloring in terms of coloring quality. In our experiments, the fixed number of colors used by the new algorithm is controlled in a centralized manner.

Graph coloring is an important tool in the study of optimization, computer science, network design, e.g., file transferring in a computer network, pattern matching, computation of Hessians matrix and so on. In this paper, we consider one important coloring, vertex coloring of a total graph, which is also called total coloring. We consider a planar graph G with maximum degree Δ(G) ≥ 8, and proved that if G contains no adjacent i, j-cycles with two chords for some i, j ∈ {5, 6, 7}, then G is total-(Δ + 1)-colorable.

The known fact that coloring of the nodes of a graph improves the performance of practical clique search algorithm is the main motivation of this paper. We will describe a number of ways in which an edge coloring scheme proposed in [8] can be used in clique search. The edge coloring provides an upper bound for the clique number. This estimate has a limitation. It will be shown that the gap between the clique number and the upper bound can be arbitrarily large. Finally, it will be shown that to determine the optimal number of colors in an edge coloring is NP-hard.

A strong edge coloring of a graph G is a proper edge coloring in which no two edges of the same color lie within distance 2 from each other. The minimum number of colors required for strong edge coloring of a graph G is called strong chromatic index and is denoted by $$\chi _{s}^{\prime } (G)$$źsź(G). Channel assignment problems are closely related with strong edge coloring problem where the colors represent frequencies. In wireless networks, assigning channels or frequencies to the links between transceivers (vertices) to avoid interference can be modelled as a strong edge coloring problem. In this paper, we determine the exact values of strong chromatic indices of interconnection networks namely butterfly network, Benes network, hypertree network and honeycomb network.

Given a graph together with a partition of its vertex set, the minimum selective coloring problem consists of selecting one vertex per partition set such that the chromatic number of the subgraph induced by the selected vertices is minimum. The contribution of this paper is twofold. First, we investigate the complexity status of the minimum selective coloring problem in some specific graph classes motivated by some models described in Demange et al. (2015). Second, we introduce a new problem that corresponds to the worst situation in the minimum selective coloring; the maximum selective coloring problem aims to select one vertex per partition set such that the chromatic number of the subgraph induced by the selected vertices is maximum. We motivate this problem by different models and give some first results concerning its complexity. (C) 2015 Elsevier B.V. All rights reserved.

We study the weighted improper coloring problem, a generalization of defective coloring. We present some hardness results and in particular we show that weighted improper coloring is not fixed-parameter tractable when parameterized by pathwidth. We generalize bounds for defective coloring to weighted improper coloring and give a bound for weighted improper coloring in terms of the sum of edge weights. Finally we give fixed-parameter algorithms for weighted improper coloring both when parameterized by treewidth and maximum degree and when parameterized by treewidth and precision of edge weights. In particular, we obtain a linear-time algorithm for weighted improper coloring of interval graphs of bounded degree.

In this paper, a new vertex-coloring problem of a fuzzy graph with crisp vertices and fuzzy edges is studied. Membership degree of a fuzzy edge is interpreted as incompatibility degree of its associated incident vertices. This interpretation can be used to define the concept of total incompatibility. Here, unlike the traditional graph coloring problems, two adjacent vertices can receive same colors; these type of vertices and their associated edge are named incompatible vertices and incompatible edge, respectively. In proposed coloring methodology, the total incompatibility of a vertex-coloring is defined as the sum of incompatibility degrees of all incompatible edges. Then, based on the minimum possible degree of total incompatibilities, fuzzy chromatic number of a fuzzy graph is introduced. In order to find an optimal k-coloring, with minimum degree of total incompatibly, firstly a binary programming problem is formulated. Then, a hybrid local search genetic algorithm is designed to solve the large-size problems. Practical uses of the proposed algorithm are illustrated and analyzed by different-size problems. Finally, a cell site assignment problem, as a real world application of the presented fuzzy graph vertex-coloring, is formulated and solved.

A unique-maximum k-coloring with respect to faces of a plane graph G is a coloring with colors 1, . . . , k so that, for each face of G, the maximum color occurs exactly once on the vertices of α. We prove that any plane graph is unique-maximum 3-colorable and has a proper unique-maximum coloring with 6 colors.

In this paper we propose a hybrid Ant Colony Optimization (ACO) algorithm for the Partition Graph Coloring Problem (PGCP). Given an undirected graph G = ( V , E ) , whose nodes are partitioned into a given number of the sets, the goal of the PGCP is to find a subset V ? ? V that contains exactly one node from each cluster and a coloring for V ? so that in the graph induced by V ? , two adjacent nodes have different colors and the total number of used colors is minimal. Our hybrid algorithm is obtained by executing a local search procedure after every ACO iteration. The performance of our algorithm is evaluated on a set of instances commonly used as benchmark and the computational results show that compared to state-of-the-art algorithms, our improved hybrid ACO algorithm achieves solid results in very short run-times.

Graph coloring is a well-known and well-studied area of graph theory that has many applications. In this dissertation, we look at two generalizations of graph coloring known as list-coloring and sum-list-coloring. In both of these types of colorings, one seeks to first assign palettes of colors to vertices and then choose a color from the corresponding palette for each vertex so that a proper coloring is obtained. A celebrated result of Thomassen states that every planar graph can be properly colored from any arbitrarily assigned palettes of five colors. This result is known as 5-list-colorability of planar graphs. Albertson asked whether Thomassen’s theorem can be extended by precoloring some vertices which are at a large enough distance apart. Hutchinson asked whether Thomassen’s theorem can be extended by allowing certain vertices to have palettes of size less than five assigned to them. In this dissertation, we explore both of these questions and answer them in the affirmative for various classes of graphs. We also provide a catalog of small configurations with palettes of different prescribed sizes and determine whether or not they can always be colored from palettes of such sizes. These small configurations can be useful in reducing certain planar graphs to obtain more information about their structure. Additionally, we look at the newer notion of sum-list-coloring where the sum choice number χSC is the parameter of interest. In sum-list-coloring, we seek to minimize the sum of varying sizes of palettes of colors assigned the vertices of a graph. We compute χSC for all graphs on at most five vertices, present some general results about sum-list-coloring, and determine χSC for certain graphs made up of cycles.

A coloring of the vertices of a graph is called convex if each subgraph induced by all vertices of the same color is connected. We consider three variants of recoloring a colored graph with minimal cost such that the resulting coloring is convex. Two variants of the problem are shown to be NP-hard on trees even if in the initial coloring each color is used to color only a bounded number of vertices. For graphs of bounded treewidth, we present a polynomial-time (2+@e)-approximation algorithm for these two variants and a polynomial-time algorithm for the third variant. Our results also show that, unless NP@?DTIME(n^O^(^l^o^g^l^o^g^n^)), there is no polynomial-time approximation algorithm with a ratio of size (1-o(1))lnlnN for the following problem: given pairs of vertices in an undirected N-vertex graph of bounded treewidth, determine the minimal possible number l for which all except l pairs can be connected by disjoint paths.

A neighbor sum distinguishing edge- k -coloring, or nsd- k -coloring for short, of a graph G is a proper edge coloring of G with elements from { 1 , 2 , ? , k } such that no pair of adjacent vertices meets the same sum of colors of G . The definition of this coloring makes sense for graphs containing no isolated edges (we call such graphs normal). Let mad ( G ) and Δ ( G ) be the maximum average degree and the maximum degree of a graph G , respectively. In this paper, we prove that every normal graph with Δ ( G ) ? 5 and mad ( G ) < 3 admits an nsd- ( Δ ( G ) + 2 ) -coloring. Our approach is based on the Combinatorial Nullstellensatz and the discharging method.

An acyclic coloring of a graph G is a proper vertex coloring such that G contains no bicolored cycles. The more restricted notion of star coloring of G is an acyclic coloring in which each path of length 3 is not bicolored. In this paper, we mainly study on the acyclic and star coloring of P4-reducible and P4-sparse graphs. Moreover, we list polynomial-time algorithms for giving an optimal acyclic or star coloring of a P4-reducible or P4-sparse graph.

We investigate the dominating-c-color number,, of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and . This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].