Showing papers on "Edge coloring published in 2007"
14 Aug 2007
TL;DR: The problem of determining whether χl(G) ≤ r, the LIST CHROMATIC NUMBER problem, is solvable in linear time for every fixed treewidth bound t, and a list-based variation, LIST EQUITABLE COLORING is W[1]-hard for trees, parameterized by the number of colors on the lists.
Abstract: We study the complexity of several coloring problems on graphs, parameterized by the treewidth t of the graph: (1) The list chromatic number χl(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list Lv of colors, where each list has length at least r, there is a choice of one color from each vertex list Lv yielding a proper coloring of G. We show that the problem of determining whether χl(G) ≤ r, the LIST CHROMATIC NUMBER problem, is solvable in linear time for every fixed treewidth bound t. The method by which this is shown is new and of general applicability. (2) The LIST COLORING problem takes as input a graph G, together with an assignment to each vertex v of a set of colors Cv. The problem is to determine whether it is possible to choose a color for vertex v from the set of permitted colors Cv, for each vertex, so that the obtained coloring of G is proper. We show that this problem is W[1]-hard, parameterized by the treewidth of G. The closely related PRECOLORING EXTENSION problem is also shown to be W[1]-hard, parameterized by treewidth. (3) An equitable coloring of a graph G is a proper coloring of the vertices where the numbers of vertices having any two distinct colors differs by at most one. We show that the problem is hard for W[1], parameterized by (t, r). We also show that a list-based variation, LIST EQUITABLE COLORING is W[1]-hard for trees, parameterized by the number of colors on the lists.
129 citations
TL;DR: In this paper, the authors introduced a new parameter k*, which is the maximum number of vertices that two faces of a graph can have in common, and proved that χc ≤ max {Δ* + 3,k* + 2, Δ* + 14, 3, k* + 6, 18}.
Abstract: 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 χc. Let Δ* be the maximum face degree of a graph. There exist plane graphs with χc = 3-2 Δ*. Ore and Plummer [5] proved that χc ≤ 2, Δ*, which bound was improved to 9-5, Δ* by Borodin, Sanders, and Zhao [1], and to 5-3,Δ* 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 χc ≤ max {Δ* + 3,k* + 2, Δ* + 14, 3, k* + 6, 18}, and if Δ* ≥ 4 and k* ≥ 4, then χc ≤ Δ* + 3,k* + 2. © 2006 Wiley Periodicals, Inc. J Graph Theory
Research for this article was done during visits of OVB, AG, and JvdH to the University of Twente, while HJB was employed there.
114 citations
TL;DR: In this article, the problem of finding the minimum number of colors needed to color a graph is reduced to finding an f-coloring of a multigraph, which is called X2f(G).
Abstract: An f-coloring of a multigraph G is a coloring of the edges of E such that each color appears at each vertex v e 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 X2f(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 Δf(G) or Δf(G)+ 1, where Δf(G) = max vev{ d(v)/f(v) } and d(v)is the degree of vertex v. A connected graph G is called f-critical if X'f(G)=Δf(G)+1 and X'f(G)=Δf(G-e) < X'f(G) for any edge e e E. Some results about f-critical graphs are given. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(3), 197202 2007
102 citations
TL;DR: In this paper, it was shown that f(2k, k2, 3) = PKk,k(3) and Kk is the only extremal graph.
Abstract: Let ℱ2k, k2 consist of all simple graphs on 2k vertices and k2 edges. For a simple graph G and a positive integer λ, let PG(λ) denote the number of proper vertex colorings of G in at most λ colors, and let f(2k, k2,λ) = max {PG(λ):Geℱ2k,k2}. We prove that f(2k, k2, 3) = PKk,k(3) and Kk,k is the only extremal graph. We also prove that f(2k, k2, 4) = (6+o(1))4k as k ➝ ∞. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 135148, 2007
92 citations
TL;DR: In this article, it was shown that the chromatic number of a planar graph with maximum degree 10 is 11, which is the same as the number of vertices in the planar graphs of the graph.
Abstract: In this article we prove that the total chromatic number of a planar graph with maximum degree 10 is 11. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 91–102, 2007
82 citations
TL;DR: Freedman, Lovasz and Schrijver as discussed by the authors proved that a graph parameter is edge reflection positive and multiplicative if and only if it can be represented by an edge coloring model.
Abstract: Solving a conjecture of M. H. Freedman, L. Lovasz and A. Schrijver, we prove that a graph parameter is edge reflection positive and multiplicative if and only if it can be represented by an edge coloring model.
58 citations
1 Jan 2007
TL;DR: In this research, the application of Iterated Local Search to the graph coloring problem is explored, different perturbation schemes are investigated and computational results on some hard instances from the DIMACS benchmark suite are presented.
Abstract: Graph coloring is a well known problem from graph theory that, when solving it with local search algorithms, is typically treated as a series of constraint satisfaction problems: for a given number of colors k, one has to find a feasible coloring; once such a coloring is found, the number of colors is decreased and the local search starts again. Here we explore the application of Iterated Local Search to the graph coloring problem. Iterated Local Search is a simple and powerful metaheuristic that has shown very good results for a variety of optimization problems. In our research we investigate different perturbation schemes and present computational results on some hard instances from the DIMACS benchmark suite.
55 citations
TL;DR: It is proved 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 multicolor spanning tree T' of Kn such that T and T' are edge-disjoint.
Abstract: 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 $2 \leq a_{1} \leq a_{2} \leq \cdots \leq a_{k} \leq (n+ 1)/ 2$, 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 $k \geq {{{n}-{2}} \choose {2}}+{3}$, then there exist two edge-disjoint multicolored spanning trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 221–232, 2007
AMS (2000) Subject Classification : 05B35, 05C05, 05C15, 05D15.
51 citations
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TL;DR: This work considers a station in which several trains might stop at the same track at thesame time and gives an optimal O(nlogn) coloring algorithm for the so called circular arc containment graphs.
Abstract: We consider a station in which several trains might stop at the same track at the same time. The trains might enter and leave the station from both sides, but the arrival and departure times and directions are fixed according to a given time table. The problem is to assign tracks to the trains such that they can enter and leave the station on time without being blocked by any other train. We consider some variation of the problem on linear time tables as well as on cyclic time tables and show how to solve them as a graph coloring problem on special graph classes. One of these classes are the so called circular arc containment graphs for which we give an optimal O(nlogn) coloring algorithm.
49 citations
TL;DR: In this paper, it was shown that at most 12 colors are necessary and sufficient for planar graphs with square-free color sequences and at most 7 colors for outerplanar graphs.
Abstract: A sequence of symbols a 1 , a 2 … is called square-free if it does not contain a subsequence of consecutive terms of the form x 1 , …, x m , x 1 , …, x m . A century ago Thue showed that there exist arbitrarily long square-free sequences using only three symbols. Sequences can be thought of as colors on the vertices of a path. Following the paper of Alon, Grytczuk, Haluszczak and Riordan, we examine graph colorings for which the color sequence is square-free on any path. The main result is that the vertices of any k -tree have a coloring of this kind using O ( c k ) colors if c > 6. Alon et al. conjectured that a fixed number of colors suffices for any planar graph. We support this conjecture by showing that this number is at most 12 for outerplanar graphs. On the other hand we prove that some outerplanar graphs require at least 7 colors. Using this latter we construct planar graphs, for which at least 10 colors are necessary.
48 citations
TL;DR: A version of weighted coloring of a graph which is motivated by some types of scheduling problems, and the associated decision problems are shown to be NP-complete for bipartite graphs, for line-graphs of bipartITE graphs, and for split graphs.
Abstract: A version of weighted coloring of a graph is introduced which is motivated by some types of scheduling problems: each node v of a graph G corresponds to some operation to be processed (with a processing time w(v)), edges represent nonsimultaneity requirements (incompatibilities) We have to assign each operation to one time slot in such a way that in each time slot, all operations assigned to this slot are compatible; the length of a time slot will be the maximum of the processing times of its operations The number k of time slots to be used has to be determined as well So, we have to find a k-coloring $${\cal S}$$ = $$({S_{1},\ldots ,S_{k}})$$ of G such that w(S 1) + ?s +w(S k ) is minimized where w(S i ) = max {w(v) :v?V} Properties of optimal solutions are discussed, and complexity and approximability results are presented Heuristic methods are given for establishing some of these results The associated decision problems are shown to be NP-complete for bipartite graphs, for line-graphs of bipartite graphs, and for split graphs
TL;DR: This work combines the idea of confluent drawings with Sugiyama-style drawings in order to reduce the edge crossings in the resultant drawings, and can be extended to obtain multi-depth confluent layered drawings.
Abstract: We combine the idea of confluent drawings with Sugiyama-style drawings in order to reduce the edge crossings in the resultant drawings. Furthermore, it is easier to understand the structures of graphs from the mixed-style drawings. The basic idea is to cover a layered graph by complete bipartite subgraphs (bicliques), then replace bicliques with tree-like structures. The biclique cover problem is reduced to a special edge-coloring problem and solved by heuristic coloring algorithms. Our method can be extended to obtain multi-depth confluent layered drawings.
TL;DR: A recently developed class of algorithms that solve global problems in unit distance wireless networks by means of local algorithms for obtaining planar subnetworks, approximations to minimum weight spanning trees, Delaunay triangulations, and relative neighbor graphs are reviewed.
TL;DR: In this paper, it was shown that the chromatic index of a G-decomposition is the minimum number of colors required to color the parts of the decomposition so that two parts which share a node get different colors.
Abstract: If G is any graph, a G-decomposition of a host graph H = (V, E) is a partition of the edge set of H into subgraphs of H which are isomorphic to G. The chromatic index of a G-decomposition is the minimum number of colors required to color the parts of the decomposition so that two parts which share a node get different colors. The G-spectrum of H is the set of all chromatic indices taken on by G-decompositions of H. If both S and T are trees, then the S-spectrum of T consists of a single value which can be computed in polynomial time. On the other hand, for any fixed tree S, not a single edge, there is a unicyclic host whose S-spectrum has two values, and if the host is allowed to be arbitrary, the S-spectrum can take on arbitrarily many values. Moreover, deciding if an integer k is in the S-spectrum of a general bipartite graph is NP-hard. We show that if G has c > 1 components, then there is a host H whose G-spectrum contains both 3 and 2c + 1. If G is a forest, then there is a tree T whose G-spectrum contains both 2 and 2c. Furthermore, we determine the complete spectra of both paths and cycles with respect to matchings. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 83–104, 2007
11 Jun 2007
TL;DR: In this paper, the mixing time of the Glauber dynamics for planar graphs with maximum degree δ was shown to be polynomial when k = Ω(δ/log δ).
Abstract: We study Markov chains for randomly sampling k-colorings of a graph with maximum degree δ. Our main result is a polynomial upper bound on the mixing time of the single-site update chain knownas the Glauber dynamics for planar graphs when k=Ω(δ/logδ). Our results can be partially extended to the more general case where the maximum eigenvalue of the adjacency matrix of the graphis at most δ1-e, for fixed e > 0.The main challenge when k ≤ δ + 1 is the possibility of "frozen" vertices, that is, vertices for which only one coloris possible, conditioned on the colors of its neighbors. Indeed, when δ = O(1), even a typical coloring canhave a constant fraction of the vertices frozen.Our proofs rely on recent advances in techniquesfor bounding mixing time using "local uniformity" properties.
TL;DR: In this article, it was shown that the problem of determining whether a bipartite graph has interval t-coloring is NP-complete in the case of @a,@b = 6, @b = 3 and t = 6.
1 Jan 2007
TL;DR: This work proposes column generation to implicitly optimize the linear programming relaxation of an independent set formulation (where there is one variable for each independent set in the graph) for graph multi-coloring.
Abstract: We present a branch-and-price framework for solving the graph multi-coloring problem. We propose column generation to implicitly optimize the linear programming relaxation of an independent set formulation (where there is one variable for each independent set in the graph) for graph multi-coloring. This approach, while requiring the solution of a difficult subproblem, is a promising method to obtain good solutions for small to moderate size problems quickly. Some implementation details and initial computational experience are presented.
TL;DR: It is proved that the harmonious coloring problem is NP-complete for connected interval and permutation graphs.
16 Jul 2007
TL;DR: This work investigates the r-COMPONENT CONNECTED COLORING COMPLETION (r-CCC) problem, that takes as input a partially colored graph, having k uncolored vertices, and asks whether the partial coloring can be completed to an r-component connected coloring.
Abstract: An r-component connected coloring of a graph is a coloring of the vertices so that each color class induces a subgraph having at most r connected components. The concept has been well-studied for r = 1, in the case of trees, under the rubric of convex coloring, used in modeling perfect phylogenies. Several applications in bioinformatics of connected coloring problems on general graphs are discussed, including analysis of protein-protein interaction networks and protein structure graphs, and of phylogenetic relationships modeled by splits trees. We investigate the r-COMPONENT CONNECTED COLORING COMPLETION (r-CCC) problem, that takes as input a partially colored graph, having k uncolored vertices, and asks whether the partial coloring can be completed to an r-component connected coloring. For r = 1 this problem is shown to be NPhard, but fixed-parameter tractable when parameterized by the number of uncolored vertices, solvable in time O*(8k). We also show that the 1-CCC problem, parameterized (only) by the treewidth t of the graph, is fixed-parameter tractable; we show this by a method that is of independent interest. The r-CCC problem is shown to be W[1]-hard, when parameterized by the treewidth bound t, for any r = 2. Our proof also shows that the problem is NP-complete for r = 2, for general graphs.
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TL;DR: It is proved that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2,4,6,8}.
Abstract: An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2,4,6,8}. We provide sufficient conditions for the existence of such a subgraph.
15 May 2007
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13 Jun 2007
TL;DR: Two new SPM allocation algorithms can outperform graph coloring when their interference graphs are superperfect or nearly so although graph coloring is admittedly more general and may also be effective to applications with arbitrary interference graphs.
Abstract: Existing methods place data or code in scratchpad memory, ie, SPM by either relying on heuristics or resorting to integer programming or mapping it to a graph coloring problemIn this work, the SPM allocation problem is formulated as an interval coloring problem The key observation is that in many embedded applications, arrays (including structs as a special case) are often related in the following way: For any two arrays, their live ranges are often such that one is either disjoint from or contains the other As a result, array interference graphs are often superperfect graphs and optimal interval colorings for such array interference graphs are possible This has led to the development of two new SPM allocation algorithms While differing in whether live range splits and spills are done sequentially or together, both algorithms place arrays in SPM based on examining the cliques in an interference graph In both cases, we guarantee optimally that all arrays in an interference graph can be placed in SPM if its size is no smaller than the clique number of the graph In the case that the SPM is not large enough, we rely on heuristics to split or spill a live range until the graph is colorable Our experiment results using embedded benchmarks show that our algorithms can outperform graph coloring when their interference graphs are superperfect or nearly so although graph coloring is admittedly more general and may also be effective to applications with arbitrary interference graphs
TL;DR: First results on @g"r","s","t(G) such as general bounds and also exact values, for example if min{r,s,t}=0 or if G is a complete graph are presented.
TL;DR: It is shown that mixed hypergraphs can be used to efficiently model several graph coloring problems including homomorphisms of simple graphs and multigraphs, circular colorings, and H,C,= graphs.
11 Oct 2007
TL;DR: A general framework which allows to convert approximation algorithms for standard node coloring into algorithms for max coloring of perfect graphs, improving and simplifying a recent result of Pemmaraju and Raman.
Abstract: We consider max coloring on hereditary graph classes. The problem is defined as follows. Given a graph G = (V,E) and positive node weights w : V → (1,∞), the goal is to find a proper node coloring of G whose color classes C1,C2,...,Ck minimize σi=1k maxυ∈Ci w(v). We design a general framework which allows to convert approximation algorithms for standard node coloring into algorithms for max coloring. The approximation ratio increases by a multiplicative factor of at most e for deterministic offline algorithms and for randomized online algorithms, and by a multiplicative factor of at most 4 for deterministic online algorithms. We consider two specific hereditary classes which are interval graphs and perfect graphs.
For interval graphs, we study the problem in several online environments. In the List Model, intervals arrive one by one, in some order. In the Time Model, intervals arrive one by one, sorted by their left endpoint. For the List Model we design a deterministic 12-competitive algorithm, a randomized 3e-competitive algorithm, and prove a lower bound of 4 on the (deterministic or randomized) competitive ratio. For the Time Model, we use simplified versions of the algorithm and the lower bound of the List Model, to achieve a deterministic 4-competitive algorithm, a randomized e-competitive algorithm, and lower bounds of φ ≅ 1.618 on the deterministic competitive ratio and 4/3 on the randomized competitive ratio. The former lower bounds hold even for unit intervals. For unit intervals in the List Model, we obtain a deterministic 8-competitive algorithm, a randomized 2e-competitive algorithm and lower bounds of 2 on the deterministic competitive ratio and 11/6 ≅ 1.8333 on the randomized competitive ratio.
Finally, we employ our framework to obtain an offline e-approximation algorithm for max coloring of perfect graphs, improving and simplifying a recent result of Pemmaraju and Raman.
TL;DR: In this paper, it was shown that the average degree of an edge-Δ-critical graph is at least 2/3(Δ+1) if Δ ≥ 2, at least 1/3 δ+1 if Δ≥ 8, and at least δ + 2 if Δ > 15.
Abstract: A graph G with maximum degree Δ and edge chromatic number X'(G) > Δ is edge-Δ-critical if X'(G-e)=Δ for every edge e of G. It is proved that the average degree of an edge-Δ-critical graph is at least 2/3(Δ+1) if Δ ≥2, at least 2/3(Δ+1) if Δ≥8, and at least 2/3(Δ+2) if Δ≥15. For large Δ, this improves on the best bound previously known, which was roughly 1/2(Δ+√2Δ). © Wiley Periodicals, Inc. J. Graph Theory 56: 194218, 2007
TL;DR: Upper bounds for the Grundy number of graphs in terms of vertex degrees, girth, clique partition number and for the line graphs are given.
13 Jun 2007
TL;DR: A model of graph related formulae that is the best possible in terms of the number of occurrences of every variable in a formula, and every property of Boolean strings that can be represented by a read-twice CNF formula is testable.
Abstract: We study a model of graph related formulae that we call the constraint-graph model A constraint-graph is a labeled multi-graph (a graph where loops and parallel edges are allowed), where each edge e is labeled by a distinct Boolean variable and every vertex is associated with a Boolean function over the variables that label its adjacent edges A Boolean assignment to the variables satisfies the constraint graph if it satisfies every vertex function We associate with a constraint-graph G the property that consists of all assignments satisfying G, denoted SAT(G) We show that the above model is quite general That is, for every property of strings P there exists a property of constraint-graphs PG such that P is testable using q queries if and only if PG is thus testable In addition, we present a large family of constraint-graphs for which SAT(G) is testable with constant number of queries As an implication of this, we infer the testability of some edge coloring problems (eg the property of two coloring of the edges in which every node is adjacent to at least one vertex of each color) Another implication is that every property of Boolean strings that can be represented by a read-twice CNF formula is testable We note that this is the best possible in terms of the number of occurrences of every variable in a formula
1 Jan 2007
TL;DR: It is proved that for sufficiently large n, in any optimal edge coloring of Kn, a random Hamilton cycle has approximately (1 − e−1)n different colors.
Abstract: In this paper we consider optimal edge colored complete graphs. We show that in any optimal edge coloring of the complete graph Kn, there is a Hamilton cycle with at most √ 8n different colors. Also we prove that in every proper edge coloring of the complete graph Kn, there is a rainbow cycle with at least n/2−1 colors (A rainbow cycle is a cycle whose all edges have different colors). We prove that for sufficiently large n, in any optimal edge coloring of Kn, a random Hamilton cycle has approximately (1 − e−1)n different colors. Finally it is proved that if using an abelian group G, we properly edge color Kn, for odd n, then it has a rainbow Hamilton cycle.
TL;DR: A random d-regular graph with d even edges, and having edges colored randomly with d/2 of each of n colors, has a rainbow Hamilton cycle with probability tending to 1 as n → ∞, for fixed d ≥ 8 as discussed by the authors.
Abstract: A rainbow subgraph of an edge-colored graph has all edges of distinct colors. A random d-regular graph with d even, and having edges colored randomly with d/2 of each of n colors, has a rainbow Hamilton cycle with probability tending to 1 as n →∞, for fixed d ≥ 8. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007