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## Coloring
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## Selected Abstracts## Coloring of Plastics, Fundamentals 2nd Ed. COLOR RESEARCH & APPLICATION, Issue 4 2004Johnny F. SuthersNo abstract is available for this article. [source] ## Concurrent Viewing of Multiple Attribute-Specific Subspaces COMPUTER GRAPHICS FORUM, Issue 3 2008Robert SisnerosAbstract In this work we present a point classification algorithm for multi-variate data. Our method is based on the concept of attribute subspaces, which are derived from a set of user specified attribute target values. Our classification approach enables users to visually distinguish regions of saliency through concurrent viewing of these subspaces in single images. We also allow a user to threshold the data according to a specified distance from attribute target values. Based on the degree of thresholding, the remaining data points are assigned radii of influence that are used for the final coloring. This limits the view to only those points that are most relevant, while maintaining a similar visual context. [source] ## Generating Animatable 3D Virtual Humans from Photographs COMPUTER GRAPHICS FORUM, Issue 3 2000WonSook LeeWe present an easy, practical and efficient full body cloning methodology. This system utilizes photos taken from the front, side and back of a person in any given imaging environment without requiring a special background or a controlled illuminating condition. A seamless generic body specified in the VRML H-Anim 1.1 format is used to generate an individualized virtual human. The system is composed of two major components: face-cloning and body-cloning. The face-cloning component uses feature points on front and side images and then applies DFFD for shape modification. Next a fully automatic seamless texture mapping is generated for 360° coloring on a 3D polygonal model. The body-cloning component has two steps: (i feature points specification, which enables automatic silhouette detection in an arbitrary background (ii two-stage body modification by using feature points and body silhouette respectively. The final integrated human model has photo-realistic animatable face, hands, feet and body. The result can be visualized in any VRML compliant browser. [source] ## Political Islam and Foreign Policy in Europe and the United States FOREIGN POLICY ANALYSIS, Issue 4 2007Elizabeth Shakman HurdThis paper is about the epistemological underpinnings of European and American foreign policy toward political Islam. European and American approaches to political Islam rely upon commonly held secular assumptions about religion and politics that have significant effects on foreign policy in Europe and the United States. Secularist epistemology produces an understanding of "normal politics" that lends a particular coloring to the politics of Muslim-majority societies. These secularist understandings affect foreign policy in two ways: first, the appearance of Islam in politics is equated with fundamentalism and intolerance, and second, the forms and degrees of separation between Islam and politics that do exist in contemporary Muslim-majority societies either do not appear at all or appear as ill-fitting imitations of a Western secular ideal. Rather than a backlash against modernity or a return to tradition, political Islam is a modern language of politics that challenges and, at times, overturns fundamental assumptions about religion and politics embedded in Western forms of secularism. [source] ## Cancer risk perceptions in an urban Mediterranean population INTERNATIONAL JOURNAL OF CANCER, Issue 1 2005Montse GarcíaAbstract The objective of our study was to analyze the perceived (belief) or adopted (behavior) measures to reduce cancer risk in a Spanish population. We used cross-sectional data from the Cornella Health Interview Survey Follow-up Study (CHIS.FU). We analyzed 1,438 subjects who in 2002 answered questions about risk perceptions on cancer and related behavior (668 males and 770 females). The benefits of avoiding cigarette smoking (95.8%), sunlight exposure (94.9%) and alcohol (81.0%) were widely recognized. On the other hand, electromagnetic fields (92.1%), food coloring and other food additives (78.4%) or pesticides (69.4%), whose role in cancer occurrence, if any, remain unproven, were clearly considered as cancer risk factors in this population. Compared to men, women more frequently reported healthy behaviors, and the role of exogenous factors (i.e., environmental risk factors) were widely popular. There was a socioeconomic gradient on cancer risk perception with respect to several lifestyle or dietary factors. Individuals with higher educational level scored lower in several risk factors than those with primary or less than primary school education. Smokers reported adopting fewer healthy behaviors than former or never smokers. How people perceive health issues and risk or make choices about their own behavior does not always follow a predictable or rational pattern. © 2005 Wiley-Liss, Inc. [source] ## Copper and calcium uptake in colored hair INTERNATIONAL JOURNAL OF COSMETIC SCIENCE, Issue 2 2010K. E. SmartJ. Cosmet. Sci., 60, 337,345 (May/June 2009) Accepted for publication December 29, 2008. Synopsis During hair coloring a number of disulfide bonds in cystine are oxidized (1) to create cysteic acid, forming binding sites for metal ions such as Ca2+ and Cu2+ from tap water (2). The increased uptake of these metals can have a detrimental impact on fiber properties,for example, reducing shine and causing a poor wet and dry feel (3). In addition, the increased uptake of copper can also contribute to further fiber damage during subsequent coloring due to its ability to take part in metal-induced radical chemistry (4). It is important to know where in the fibers these metals are located in order to either effectively remove these metals or control their chemistry. Nanoscale secondary ion mass spectrometry (NanoSIMS) has been used to locate the calcium and copper within hair that has been treated with a colorant and washed multiple times in tap water containing these ions. Untreated hair is used as a baseline standard material. Images with up to 50-nm spatial resolution of the preferential locations of calcium uptake were obtained, showing a high concentration of calcium in the cuticle region of colored hair, specifically in the sulfur-rich regions (A-layer and exocuticle). [source] ## Process Optimization and Consumer Acceptability of Salted Ground Beef Patties Cooked and Held Hot in Flavored Marinade JOURNAL OF FOOD SCIENCE, Issue 7 2010Subash ShresthaAbstract:, Food safety is paramount for cooking hamburger. The center must reach 71 °C (or 68 °C for 15 s) to assure destruction of,E. coli,O157:H7 and other food pathogens. This is difficult to achieve during grilling or frying of thick burgers without overcooking the surface. Thus, the feasibility of partially or completely cooking frozen patties in liquid (93 °C water) together with hot holding in liquid was investigated. Initial studies demonstrated that compared to frying, liquid cooking decreased (P,< 0.05) patty diameter (98 compared with 93 mm) and increased (P,< 0.05) thickness (18.1 compared with 15.6 mm). Liquid cooked patties had greater weight loss (P,< 0.05) immediately after cooking (29 compared with 21%), but reabsorbed moisture and were not different from fried patties after 1 h hot water holding (61 °C). Protein and fat content were not affected by cooking method. However, liquid cooked patties were rated lower (P,< 0.05) than fried patties for appearance (5.7 compared with 7.5) and flavor (5.9 compared with 7.5). An 8-member focus group then evaluated methods to improve both appearance and flavor. Salted, grill-marked patties were preferred, and caramel coloring was needed in the marinade to obtain acceptable flavor and color during liquid cooking or hot holding. Patties with 0.75% salt that were grill-marked and then finish-cooked in hot marinade (0.75% salt, 0.3% caramel color) were rated acceptable (P,< 0.05) by consumers for up to 4 h hot holding in marinade, with mean hedonic panel ratings > 7.0 (like moderately) for appearance, juiciness, flavor, and texture. Practical Application: Grill-marked and marinade-cooked ground beef patties reached a safe internal cooking temperature without overcooking the surface. Burgers cooked using this method maintained high consumer acceptability right after cooking and for up to 4 h of hot holding. Consumers and foodservice operations could use this method without specialized equipment, and instead use inexpensive and common equipment such as a soup pot or a restaurant steam table. Use of marinades (salt/caramel color or others) in this cooking and holding method provides a nearly endless culinary flavoring opportunity. [source] ## Influence of Exposure to Light on the Sensorial Quality of Minimally Processed Cauliflower JOURNAL OF FOOD SCIENCE, Issue 1 2007Susana Sanz CerveraABSTRACT:, The impact of lighting on minimally processed cauliflower packaged in 4 different film types (PVC and 3 P-Plus) has been measured and quantified. The effect on the sensorial quality of storage at 4 °C in darkness and partial or continuous lighting was evaluated. The gas concentrations in the packages and the weight losses were also determined. Atmosphere composition inside the packages depended on both the permeability of the film used for the packaging and exposure to light. Samples stored with lighting maintained the gaseous exchange between plant tissue and the atmosphere inside the packages for longer periods than in samples kept in darkness. This prompted a greater loss of water vapor as well as the development of atmospheres with low levels of O2 and high levels of CO2 in the samples packed with less permeable films. The most important aspect in sensory evaluation was color. In instrumental color evaluation, coordinates h* and L* were the main means for estimating color evolution. The presence of light accelerated browning in the cut zones. The development of abnormal coloring in these areas marked the end of shelf life for minimally processed cauliflower. Among the sensory attributes studied, color was the most affected by exposure to light. Samples packed in P-Plus 120 film displayed the lowest level of color deterioration in the cut zones. However, under lit conditions, the low permeability of this film caused atmospheres with very low O2 contents and high CO2 contents. These atmospheres produced a loss of texture and the development of off-odors. [source] ## Subgraph-avoiding coloring of graphs JOURNAL OF GRAPH THEORY, Issue 4 2010Jia ShenAbstract Given a "forbidden graph" F and an integer k, an F-avoiding k-coloring of a graph G is a k -coloring of the vertices of G such that no maximal F -free subgraph of G is monochromatic. The F-avoiding chromatic numberacF(G) is the smallest integer k such that G is F -avoiding k -colorable. In this paper, we will give a complete answer to the following question: for which graph F, does there exist a constant C, depending only on F, such that acF(G) , C for any graph G? For those graphs F with unbounded avoiding chromatic number, upper bounds for acF(G) in terms of various invariants of G are also given. Particularly, we prove that , where n is the order of G and F is not Kk or . © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 300,310, 2010 [source] ## Multi-coloring the Mycielskian of graphs JOURNAL OF GRAPH THEORY, Issue 4 2010Wensong LinAbstract A k -fold coloring of a graph is a function that assigns to each vertex a set of k colors, so that the color sets assigned to adjacent vertices are disjoint. The kth chromatic number of a graph G, denoted by ,k(G), is the minimum total number of colors used in a k -fold coloring of G. Let µ(G) denote the Mycielskian of G. For any positive integer k, it holds that ,k(G) + 1,,k(µ(G)),,k(G) + k (W. Lin, Disc. Math., 308 (2008), 3565,3573). Although both bounds are attainable, it was proved in (Z. Pan, X. Zhu, Multiple coloring of cone graphs, manuscript, 2006) that if k,2 and ,k(G),3k,2, then the upper bound can be reduced by 1, i.e., ,k(µ(G)),,k(G) + k,1. We conjecture that for any n,3k,1, there is a graph G with ,k(G)=n and ,k(µ(G))=n+ k. This is equivalent to conjecturing that the equality ,k(µ(K(n, k)))=n+k holds for Kneser graphs K(n, k) with n,3k,1. We confirm this conjecture for k=2, 3, or when n is a multiple of k or n,3k2/ln k. Moreover, we determine the values of ,k(µ(C2q+1)) for 1,k,q. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 311,323, 2010 [source] ## Star coloring planar graphs from small lists JOURNAL OF GRAPH THEORY, Issue 4 2010André KündgenAbstract A star coloring of a graph is a proper vertex-coloring such that no path on four vertices is 2-colored. We prove that the vertices of every planar graph of girth 6 (respectively 7, 8) can be star colored from lists of size 8 (respectively 7, 6). We give an example of a planar graph of girth 5 that requires 6 colors to star color. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 324,337, 2010 [source] ## d -Regular graphs of acyclic chromatic index at least d+2 JOURNAL OF GRAPH THEORY, Issue 3 2010Manu BasavarajuAbstract An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a,(G). It was conjectured by Alon, Sudakov and Zaks (and earlier by Fiamcik) that a,(G),,+2, where ,=,(G) denotes the maximum degree of the graph. Alon et al. also raised the question whether the complete graphs of even order are the only regular graphs which require ,+2 colors to be acyclically edge colored. In this article, using a simple counting argument we observe not only that this is not true, but in fact all d -regular graphs with 2n vertices and d>n, requires at least d+ 2 colors. We also show that a,(Kn, n),n+ 2, when n is odd using a more non-trivial argument. (Here Kn, n denotes the complete bipartite graph with n vertices on each side.) This lower bound for Kn, n can be shown to be tight for some families of complete bipartite graphs and for small values of n. We also infer that for every d, n such that d,5, n,2d+ 3 and dn even, there exist d -regular graphs which require at least d+2-colors to be acyclically edge colored. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 226,230, 2010 [source] ## Non-rainbow colorings of 3-, 4- and 5-connected plane graphs JOURNAL OF GRAPH THEORY, Issue 2 2010k DvoAbstract We study vertex-colorings of plane graphs that do not contain a rainbow face, i.e., a face with vertices of mutually distinct colors. If G is a 3 -connected plane graph with n vertices, then the number of colors in such a coloring does not exceed . If G is 4 -connected, then the number of colors is at most , and for n,3(mod8), it is at most . Finally, if G is 5 -connected, then the number of colors is at most . The bounds for 3 -connected and 4 -connected plane graphs are the best possible as we exhibit constructions of graphs with colorings matching the bounds. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 129,145, 2010 [source] ## A structure theorem for graphs with no cycle with a unique chord and its consequences JOURNAL OF GRAPH THEORY, Issue 1 2010Nicolas TrotignonAbstract We give a structural description of the class ,, of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in ,, is either in some simple basic class or has a decomposition. Basic classes are chordless cycles, cliques, bipartite graphs with one side containing only nodes of degree 2 and induced subgraphs of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for ,,, i.e. every graph in ,, can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations, and all graphs built this way are in ,,. This has several consequences: an ,,(nm) -time algorithm to decide whether a graph is in ,,, an ,,(n+ m) -time algorithm that finds a maximum clique of any graph in ,,, and an ,,(nm) -time coloring algorithm for graphs in ,,. We prove that every graph in ,, is either 3-colorable or has a coloring with , colors where , is the size of a largest clique. The problem of finding a maximum stable set for a graph in ,, is known to be NP-hard. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 31,67, 2010 [source] ## Perfect coloring and linearly ,-bound P6 -free graphs JOURNAL OF GRAPH THEORY, Issue 4 2007S. A. ChoudumAbstract We derive decomposition theorems for P6, K1 + P4 -free graphs, P5, K1 + P4 -free graphs and P5, K1 + C4 -free graphs, and deduce linear ,-binding functions for these classes of graphs (here, Pn (Cn) denotes the path (cycle) on n vertices and K1 + G denotes the graph obtained from G by adding a new vertex and joining it with every vertex of G). Using the same techniques, we also obtain an optimal ,-binding function for P5, C4 -free graphs which is an improvement over that given in [J. L. Fouquet, V. Giakoumakis, F. Maire, and H. Thuillier, 1995, Discrete Math, 146, 33,44.]. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 293,306, 2007 [source] ## Multicolored trees in complete graphs JOURNAL OF GRAPH THEORY, Issue 3 2007S. AkbariAbstract A multicolored tree is a tree whose edges have different colors. Brualdi and Hollingsworth 5 proved in any proper edge coloring of the complete graph K2n(n > 2) with 2n , 1 colors, there are two edge-disjoint multicolored spanning trees. In this paper we generalize this result showing that if (a1,,, ak) is a color distribution for the complete graph Kn, n,,,5, such that , then there exist two edge-disjoint multicolored spanning trees. Moreover, we prove that for any edge coloring of the complete graph Kn with the above distribution if T is a non-star multicolored spanning tree of Kn, then there exists a multicolored spanning tree T' of Kn such that T and T' are edge-disjoint. Also it is shown that if Kn, n , 6, is edge colored with k colors and , then there exist two edge-disjoint multicolored spanning trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 221,232, 2007 [source] ## Acyclic 5-choosability of planar graphs without small cycles JOURNAL OF GRAPH THEORY, Issue 3 2007Mickaël MontassierAbstract A proper vertex coloring of a graph G,=,(V,E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L -list colorable if for a given list assignment L,=,{L(v): v:,,,V}, there exists a proper acyclic coloring,,,of G such that ,(v),,, L(v) for all v,,, V. If G is acyclically L -list colorable for any list assignment with |L (v)|, k for all v,,, V, then G is acyclically k -choosable. In this article, we prove that every planar graph G without 4- and 5-cycles, or without 4- and 6-cycles is acyclically 5-choosable. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 245,260, 2007 [source] ## A new upper bound on the cyclic chromatic number, JOURNAL OF GRAPH THEORY, Issue 1 2007O. V. BorodinAbstract A cyclic coloring of a plane graph is a vertex coloring such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a graph is its cyclic chromatic number ,c. Let ,* be the maximum face degree of a graph. There exist plane graphs with ,c = ,3/2 ,*,. Ore and Plummer [5] proved that ,c , 2, ,*, which bound was improved to ,9/5, ,*, by Borodin, Sanders, and Zhao [1], and to ,5/3,,*, by Sanders and Zhao [7]. We introduce a new parameter k*, which is the maximum number of vertices that two faces of a graph can have in common, and prove that ,c , max {,* + 3,k* + 2, ,* + 14, 3, k* + 6, 18}, and if ,* , 4 and k* , 4, then ,c , ,* + 3,k* + 2. © 2006 Wiley Periodicals, Inc. J Graph Theory [source] ## Improved bounds for the chromatic number of a graph JOURNAL OF GRAPH THEORY, Issue 3 2004S. Louis HakimiAbstract After giving a new proof of a well-known theorem of Dirac on critical graphs, we discuss the elegant upper bounds of Matula and Szekeres-Wilf which follow from it. In order to improve these bounds, we consider the following fundamental coloring problem: given an edge-cut (V1, V2) in a graph G, together with colorings of ,V1, and ,V2,, what is the least number of colors in a coloring of G which "respects" the colorings of ,V1, and ,V2, ? We give a constructive optimal solution of this problem, and use it to derive a new upper bound for the chromatic number of a graph. As easy corollaries, we obtain several interesting bounds which also appear to be new, as well as classical bounds of Dirac and Ore, and the above mentioned bounds of Matula and Szekeres-Wilf. We conclude by considering two algorithms suggested by our results. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 217,225, 2004 [source] ## Highly stable electrochromic polyamides based on N,N -bis(4-aminophenyl)- N,,N,-bis(4- tert -butylphenyl)-1,4-phenylenediamine JOURNAL OF POLYMER SCIENCE (IN TWO SECTIONS), Issue 9 2009Sheng-Huei HsiaoAbstract A new triphenylamine-containing aromatic diamine monomer, N,N -bis(4-aminophenyl)- N,,N,-bis(4- tert -butylphenyl)-1,4-phenylenediamine, was synthesized by an established synthetic procedure from readily available reagents. A novel family of electroactive polyamides with di- tert -butyl-substituted N,N,N,,N,-tetraphenyl-1,4-phenylenediamine units were prepared via the phosphorylation polyamidation reactions of the newly synthesized diamine monomer with various aromatic or aliphatic dicarboxylic acids. All the polymers were amorphous with good solubility in many organic solvents, such as N -methyl-2-pyrrolidinone (NMP) and N,N -dimethylacetamide, and could be solution-cast into tough and flexible polymer films. The polyamides derived from aromatic dicarboxylic acids had useful levels of thermal stability, with glass-transition temperatures of 269,296 °C, 10% weight-loss temperatures in excess of 544 °C, and char yields at 800 °C in nitrogen higher than 62%. The dilute solutions of these polyamides in NMP exhibited strong absorption bands centered at 316,342 nm and photoluminescence maxima around 362,465 nm in the violet-blue region. The polyamides derived from aliphatic dicarboxylic acids were optically transparent in the visible region and fluoresced with a higher quantum yield compared with those derived from aromatic dicarboxylic acids. The hole-transporting and electrochromic properties were examined by electrochemical and spectro-electrochemical methods. Cyclic voltammograms of the polyamide films cast onto an indium-tin oxide-coated glass substrate exhibited two reversible oxidation redox couples at 0.57,0.60 V and 0.95,0.98 V versus Ag/AgCl in acetonitrile solution. The polyamide films revealed excellent elcterochemical and electrochromic stability, with a color change from a colorless or pale yellowish neutral form to green and blue oxidized forms at applied potentials ranging from 0.0 to 1.2 V. These anodically coloring polymeric materials showed interesting electrochromic properties, such as high coloration efficiency (CE = 216 cm2/C for the green coloring) and high contrast ratio of optical transmittance change (,T%) up to 64% at 424 nm and 59% at 983 nm for the green coloration, and 90% at 778 nm for the blue coloration. The electroactivity of the polymer remains intact even after cycling 500 times between its neutral and fully oxidized states. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 2330,2343, 2009 [source] ## WHEELCHAIR-MODIFIED SENSORY BOOTHS DO NOT INFLUENCE EVALUATION OF CHOCOLATE PUDDING JOURNAL OF SENSORY STUDIES, Issue 3 2002MARY ELLEN CAMIREABSTRACT The objective of this study was to demonstrate that the dimensions of a sensory booth modified to comply with the Americans with Disabilities Act of 1990 (ADA) guidelines have no impact on consumer evaluations of chocolate pudding. One set of booths each were constructed per ASTM and ADA guidelines. Control pudding was prepared according to package directions with 2% fat milk; other treatments were modified with red food coloring, sugar or nonfat milk. Twenty-four panelists evaluated instant chocolate puddings in directional difference and hedonic tests in each type of booth. Booth type and test date had no effect on judgments. [source] ## Rainbow trees in graphs and generalized connectivity NETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2010Gary ChartrandAbstract An edge-colored tree T is a rainbow tree if no two edges of T are assigned the same color. Let G be a nontrivial connected graph of order n and let k be an integer with 2 , k , n. A k -rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S , V(T). The minimum number of colors needed in a k -rainbow coloring of G is the k -rainbow index of G. For every two integers k and n , 3 with 3 , k , n, the k -rainbow index of a unicyclic graph of order n is determined. For a set S of vertices in a connected graph G of order n, a collection {T1,T2,,,T,} of trees in G is said to be internally disjoint connecting S if these trees are pairwise edge-disjoint and V(Ti) , V(Tj) = S for every pair i,j of distinct integers with 1 , i,j , ,. For an integer k with 2 , k , n, the k -connectivity ,k(G) of G is the greatest positive integer , for which G contains at least , internally disjoint trees connecting S for every set S of k vertices of G. It is shown that ,k(Kn)=n,,k/2, for every pair k,n of integers with 2 , k , n. For a nontrivial connected graph G of order n and for integers k and , with 2 , k , n and 1 , , , ,k(G), the (k,,)-rainbow index rxk,,(G) of G is the minimum number of colors needed in an edge coloring of G such that G contains at least , internally disjoint rainbow trees connecting S for every set S of k vertices of G. The numbers rxk,,(Kn) are determined for all possible values k and , when n , 6. It is also shown that for , , {1, 2}, rx3,,(Kn) = 3 for all n , 6. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010 [source] ## A branch-and-cut algorithm for partition coloring NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2010Yuri FrotaAbstract Let G = (V, E, Q) be a undirected graph, where V is the set of vertices, E is the set of edges, and Q = {Q1,,,Qq} is a partition of V into q subsets. We refer to Q1,,,Qq as the components of the partition. The partition coloring problem (PCP) consists of finding a subset V, of V with exactly one vertex from each component Q1,,,Qq and such that the chromatic number of the graph induced in G by V, is minimum. This problem is a generalization of the graph coloring problem. This work presents a branch-and-cut algorithm proposed for PCP. An integer programing formulation and valid inequalities are proposed. A tabu search heuristic is used for providing primal bounds. Computational experiments are reported for random graphs and for PCP instances originating from the problem of routing and wavelength assignment in all-optical WDM networks. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010 [source] ## Improper coloring of unit disk graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2009Frédéric HavetAbstract Motivated by a satellite communications problem, we consider a generalized coloring problem on unit disk graphs. A coloring is k -improper if no more than k neighbors of every vertex have the same colour as that assigned to the vertex. The k -improper chromatic number ,k(G) is the least number of colors needed in a k -improper coloring of a graph G. The main subject of this work is analyzing the complexity of computing ,k for the class of unit disk graphs and some related classes, e.g., hexagonal graphs and interval graphs. We show NP-completeness in many restricted cases and also provide both positive and negative approximability results. Because of the challenging nature of this topic, many seemingly simple questions remain: for example, it remains open to determine the complexity of computing ,k for unit interval graphs. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009 [source] ## Some results about f -critical graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2007Guizhen LiuAbstract An f -coloring of a multigraph G is a coloring of the edges of E such that each color appears at each vertex v , V at most f(v) times. The minimum number of colors needed to f -color G is called the f -chromatic index of G and is denoted by ,,f(G). Various scheduling problems on networks are reduced to finding an f -coloring of a multigraph. Any simple graph G has f -chromatic index equal to ,f(G) or ,f(G)+ 1, where ,f(G) = max v,V{, ,} and d(v) is the degree of vertex v. A connected graph G is called f -critical if ,,f(G)=,f(G)+1 and ,,f(G)=,f(G,e) < ,,f(G) for any edge e , E. Some results about f -critical graphs are given. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(3), 197,202 2007 [source] ## Approximate L(,1,,2,,,,t)-coloring of trees and interval graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2007Alan A. BertossiAbstract Given a vector (,1,,2,,,,t) of nonincreasing positive integers, and an undirected graph G = (V,E), an L(,1,,2,,,,t)-coloring of G is a function f from the vertex set V to a set of nonnegative integers such that ,f(u) , f(v), , ,i, if d(u,v) = i, 1 , i , t, where d(u,v) is the distance (i.e., the minimum number of edges) between the vertices u and v. An optimal L(,1,,2,,,,t)-coloring for G is one minimizing the largest integer used over all such colorings. Such a coloring problem has relevant applications in channel assignment for interference avoidance in wireless networks. This article presents efficient approximation algorithms for L(,1,,2,,,,t)-coloring of two relevant classes of graphs,trees, and interval graphs. Specifically, based on the notion of strongly simplicial vertices, O(n(t + ,1)) and O(nt2,1) time algorithms are proposed to find ,-approximate colorings on interval graphs and trees, respectively, where n is the number of vertices and , is a constant depending on t and ,1,,,,t. Moreover, an O(n) time algorithm is given for the L(,1,,2)-coloring of unit interval graphs, which provides a 3-approximation. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 49(3), 204,216 2007 [source] ## On some properties of suboptimal colorings of graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2004Ivo BlöchligerAbstract Starting from the trivial observation that, in any optimal coloring of a graph, there always exists a node v such that its neighborhood N(v) contains all colors, we examine related properties in suboptimal colorings (i.e., those using more than ,(G) colors, where ,(G) is the chromatic number). In particular, we show that, in any (,(G) + p)-coloring of G, there is a node v such that its generalized neighborhood Nq(v) with q = max{2p , 1, 2} contains ,(G) colors for p , 1. Additional properties of (,(G) + p)-colorings are also given. © 2004 Wiley Periodicals, Inc. [source] ## Social network coordination and graph routing NETWORKS: AN INTERNATIONAL JOURNAL, Issue 1 2003Shmuel OnnAbstract We consider the problem of coordinating robots moving on a network. Each robot is autonomous and needs to visit various sites of the network at various times. The sequence of destinations for each robot changes dynamically and unpredictably. Recently, Onn and Tennenholtz showed that the problem can be solved by introducing a social law on the network, which, once obeyed by all robots, enables each to move to any desired destination without collisions and regardless of the actions of other robots, needing neither central coordination nor mutual communication. This social law can be derived from a suitably defined routing of the graph underlying the network. Here, we study the complexity of routing. We provide an effective characterization of 2-routable graphs, and by establishing a correspondence between hypergraph coloring and graph routing, we show that computing or approximating an optimal routing is generally hard. We also discuss routing in planar graphs, which often underlie robotic networks and show that the correspondence between coloring and routing together with the Four Color Theorem guarantee the existence of small and effectively computable routings in bipartite planar graphs of small radius. The complexity of routing arbitrary planar graphs remains open. © 2002 Wiley Periodicals, Inc. [source] ## Solving partial constraint satisfaction problems with tree decomposition NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2002Arie M. C. A. KosterAbstract In this paper, we describe a computational study to solve hard partial constraint satisfaction problems (PCSPs) to optimality. The PCSP is a general class of problems that contains a diversity of problems, such as generalized subgraph problems, MAX-SAT, Boolean quadratic programs, and assignment problems like coloring and frequency planning. We present a dynamic programming algorithm that solves PCSPs based on the structure (tree decomposition) of the underlying constraint graph. With the use of dominance and bounding techniques, we are able to solve small and medium-size instances of the problem to optimality and to obtain good lower bounds for large-size instances within reasonable time and memory limits. © 2002 Wiley Periodicals, Inc. [source] ## Solving the minimum-weighted coloring problem NETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2001Massimiliano CaramiaAbstract Weighted coloring is a generalization of the well-known vertex (unweighted) coloring for which a number of exact algorithms have been presented in the literature. We are not aware of any optimal method specifically designed for the minimum-weighted coloring problem on arbitrary graphs. Only a few heuristics have been developed with the goal of finding tighter upper bounds for the maximum-weighted clique problem. Moreover, as shown in the paper, a straightforward reduction of a weighted instance into an unweighted one permits us to solve only very small instances. In this paper, we present a branch-and-bound algorithm for the weighted case capable of solving random graphs of up to 90 vertices for any edge density with integer weights uniformly drawn from the range [1, ,,10]. Likewise, we have used properly modified benchmark instances borrowed from vertex coloring as a further test bed for our algorithm. © 2001 John Wiley & Sons, Inc. [source] |