Showing papers on "Edge coloring published in 2014"

A Survey on the Computational Complexity of Colouring Graphs with Forbidden Subgraphs

Abstract: For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u) eq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring of $G$ exists. If $k$ is fixed (that is, it is not part of the input), we have the decision problem $k$-Colouring instead. We survey known results on the computational complexity of Colouring and $k$-Colouring for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex.

Entropy compression method applied to graph colorings

Abstract: Based on the algorithmic proof of Lovasz local lemma due to Moser and Tardos, the works of Grytczuk et al on words, and Dujmovic et al on colorings, Esperet and Parreau developed a framework to prove upper bounds for several chromatic numbers (in particular acyclic chromatic index, star chromatic number and Thue chromatic number) using the so-called \emph{entropy compression method} Inspired by this work, we propose a more general framework and a better analysis This leads to improved upper bounds on chromatic numbers and indices In particular, every graph with maximum degree $\Delta$ has an acyclic chromatic number at most $\frac{3}{2}\Delta^{\frac43} + O(\Delta)$ Also every planar graph with maximum degree $\Delta$ has a facial Thue choice number at most $\Delta + O(\Delta^\frac 12)$ and facial Thue choice index at most $10$

Closing complexity gaps for coloring problems on H-free graphs

Abstract: If a graph G contains no subgraph isomorphic to some graph H, then G is called H-free. A coloring of a graph G=(V,E) is a mapping c:V->{1,2,...} such that no two adjacent vertices have the same color, i.e., c(u) c(v) if [email protected]?E; if |c(V)|=0, the Listk-Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u) that must be a subset of {1,...,k}. We show that List 4-Coloring is NP-complete for P"6-free graphs, where P"6 is the path on six vertices. This completes the classification of Listk-Coloring for P"6-free graphs.

The condensation phase transition in random graph coloring

Abstract: Based on a non-rigorous formalism called the "cavity method", physicists have put forward intriguing predictions on phase transitions in discrete structures. One of the most remarkable ones is that in problems such as random $k$-SAT or random graph $k$-coloring, very shortly before the threshold for the existence of solutions there occurs another phase transition called "condensation" [Krzakala et al., PNAS 2007]. The existence of this phase transition appears to be intimately related to the difficulty of proving precise results on, e.g., the $k$-colorability threshold as well as to the performance of message passing algorithms. In random graph $k$-coloring, there is a precise conjecture as to the location of the condensation phase transition in terms of a distributional fixed point problem. In this paper we prove this conjecture for $k$ exceeding a certain constant $k_0$.

Online Coloring of Bipartite Graphs with and without Advice

Abstract: In the online version of the well-known graph coloring problem, the vertices appear one after the other together with the edges to the already known vertices and have to be irrevocably colored immediately after their appearance. We consider this problem on bipartite, i.e., two-colorable graphs. We prove that at least ?1.13746?log2(n)?0.49887? colors are necessary for any deterministic online algorithm to be able to color any given bipartite graph on n vertices, thus improving on the previously known lower bound of ?log2 n?+1 for sufficiently large n. Recently, the advice complexity was introduced as a method for a fine-grained analysis of the hardness of online problems. We apply this method to the online coloring problem and prove (almost) tight linear upper and lower bounds on the advice complexity of coloring a bipartite graph online optimally or using 3 colors. Moreover, we prove that $O(\sqrt{n})$ advice bits are sufficient for coloring any bipartite graph on n vertices with at most ?log2 n? colors.

On Neighbor-Distinguishing Index of Planar Graphs

Abstract: A proper edge coloring of a graph G without isolated edges is neighbor-distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The neighbor-distinguishing index of G is the minimum number ndi(G) of colors in a neighbor-distinguishing edge coloring of G. Zhang, Liu, and Wang in 2002 conjectured that if G is a connected graph of order at least 6. In this article, the conjecture is verified for planar graphs with maximum degree at least 12.

Distributed algorithms for the Lovász local lemma and graph coloring

Abstract: The Lovasz Local Lemma (LLL), introduced by Erdos and Lovasz in 1975, is a powerful tool of the probabilistic method that allows one to prove that a set of n "bad" events do not happen with non-zero probability, provided that the events have limited dependence. However, the LLL itself does not suggest how to find a point avoiding all bad events. Since the work of Beck (1991) there has been a sustained effort to find a constructive proof (i.e. an algorithm) for the LLL or weaker versions of it. In a major breakthrough Moser and Tardos (2010) showed that a point avoiding all bad events can be found efficiently. They also proposed a distributed/parallel version of their algorithm that requires O(log2 n) rounds of communication in a distributed network.In this paper we provide two new distributed algorithms for the LLL that improve on both the efficiency and simplicity of the Moser-Tardos algorithm. For clarity we express our results in terms of the symmetric LLL though both algorithms deal with the asymmetric version as well. Let p bound the probability of any bad event and d be the maximum degree in the dependency graph of the bad events. When epd2

Rainbow hamilton cycles in random graphs

Abstract: One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdi¾?s-Renyi random graph Gn,p is around p~logn+loglognn. Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3-uniform hypergraph by connecting 3-uniform hypergraphs to edge-colored graphs.

On the Facial Thue Choice Index via Entropy Compression

Abstract: A sequence is nonrepetitive if it contains no identical consecutive subsequences. An edge coloring of a path is nonrepetitive if the sequence of colors of its consecutive edges is nonrepetitive. By the celebrated construction of Thue, it is possible to generate nonrepetitive edge colorings for arbitrarily long paths using only three colors. A recent generalization of this concept implies that we may obtain such colorings even if we are forced to choose edge colors from any sequence of lists of size 4 (while sufficiency of lists of size 3 remains an open problem). As an extension of these basic ideas, Havet, Jendrol', Sotak, and Skrabul'akova proved that for each plane graph, eight colors are sufficient to provide an edge coloring so that every facial path is nonrepetitively colored. In this article, we prove that the same is possible from lists, provided that these have size at least 12. We thus improve the previous bound of 291 (proved by means of the Lovasz Local Lemma). Our approach is based on the Moser–Tardos entropy-compression method and its recent extensions by Grytczuk, Kozik, and Micek, and by Dujmovic, Joret, Kozik, and Wood.

Classes of graphs critical for the edge list-ranking problem

Abstract: The edge list-ranking problem is a generalization of the classical edge coloring problem, and it is a mathematical model for some parallel processes. The computational complexity of this problem is under study for graph sets closed under isomorphism and deletion of vertices (hereditary classes). All finitely defined and minor-closed cases are described for which the problem is polynomial-time solvable (unless N=NP).We find the whole set of “critical” graph classes whose inclusion in a finitely defined class is equivalent to intractability of the edge list-ranking problem in this class (unless N=NP). It seems to be the first result on a complete description for nonartificial NP-complete graph problems. For this problem, we prove constructively that, among the inclusion minimal NP-complete hereditary cases, there are exactly five finitely defined classes and the only minor-closed class.

On the Algorithmic Lov\'asz Local Lemma and Acyclic Edge Coloring

Abstract: The algorithm for Lov\'{a}sz Local Lemma by Moser and Tardos gives a constructive way to prove the existence of combinatorial objects that satisfy a system of constraints. We present an alternative probabilistic analysis of the algorithm that does not involve reconstructing the history of the algorithm. We apply our technique to improve the best known upper bound to acyclic chromatic index. Specifically we show that a graph with maximum degree $\Delta$ has an acyclic proper edge coloring with at most $\lceil 3.74(\Delta-1)\rceil+1 $ colors, whereas the previously known best bound was $4(\Delta-1)$. The same technique is also applied to improve corresponding bounds for graphs with bounded girth. An interesting aspect of the latter application is that the probability of the "undesirable" events do not have a uniform upper bound, i.e., it constitutes a case of the asymmetric Lov\'{a}sz Local Lemma.

Acyclic Edge Coloring through the Lov\'asz Local Lemma

Abstract: We give a probabilistic analysis of a Moser-type algorithm for the Lovasz Local Lemma (LLL), adjusted to search for acyclic edge colorings of a graph. We thus improve the best known upper bound to acyclic chromatic index, also obtained by analyzing a similar algorithm, but through the entropic method (basically counting argument). Specifically we show that a graph with maximum degree $\Delta$ has an acyclic proper edge coloring with at most $\lceil 3.74(\Delta-1)\rceil+1 $ colors, whereas, previously, the best bound was $4(\Delta-1)$. The main contribution of this work is that it comprises a probabilistic analysis of a Moser-type algorithm applied to events pertaining to dependent variables.

Strong Chromatic Index Of Planar Graphs With Large Girth

Abstract: Let Δ ≥ 4 be an integer. In this note, we prove that every planar graph with maximum degree Δ and girth at least 1 Δ+46 is strong (2Δ−1)-edgecolorable, that is best possible (in terms of number of colors) as soon as G contains two adjacent vertices of degree Δ. This improves [6] when Δ ≥ 6.

Interval Non-edge-Colorable Bipartite Graphs and Multigraphs

Abstract: An edge-coloring of a graph G with colors 1,...,t is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In 1991, Erdi¾?s constructed a bipartite graph with 27 vertices and maximum degree 13 that has no interval coloring. Erdi¾?s's counterexample is the smallest in a sense of maximum degree known bipartite graph that is not interval colorable. On the other hand, in 1992, Hansen showed that all bipartite graphs with maximum degree at most 3 have an interval coloring. In this article, we give some methods for constructing of interval non-edge-colorable bipartite graphs. In particular, by these methods, we construct three bipartite graphs that have no interval coloring, contain 20, 19, 21 vertices and have maximum degree 11, 12, 13, respectively. This partially answers a question that arose in [T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar problems for bipartite multigraphs.

Coloring graphs using topology.

Abstract: Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if all interior edge degrees are even. Using a refinement process which cuts the interior along surfaces we can adapt the degrees along the boundary of that surface. More efficient is a self-cobordism of G with itself with a host graph discretizing the product of G with an interval. It follows from the fact that Euler curvature is zero everywhere for three dimensional geometric graphs, that the odd degree edge set O is a cycle and so a boundary if H is simply connected. A reduction to minimal colouring would imply the four colour theorem. The method is expected to give a reason "why 4 colours suffice" and suggests that every two dimensional geometric graph of arbitrary degree and orientation can be coloured by 5 colours: since the projective plane can not be a boundary of a 3-dimensional graph and because for higher genus surfaces, the interior H is not simply connected, we need in general to embed a surface into a 4-dimensional simply connected graph in order to colour it. This explains the appearance of the chromatic number 5 for higher degree or non-orientable situations, a number we believe to be the upper limit. For every surface type, we construct examples with chromatic number 3,4 or 5, where the construction of surfaces with chromatic number 5 is based on a method of Fisk. We have implemented and illustrated all the topological aspects described in this paper on a computer. So far we still need human guidance or simulated annealing to do the refinements in the higher dimensional host graph.

On Cubic Bridgeless Graphs Whose Edge-Set Cannot be Covered by Four Perfect Matchings

Abstract: The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this article, we prove that deciding whether this number is at most four for a given cubic bridgeless graph is NP-complete. We also construct an infinite family F of snarks cyclically 4-edge-connected cubic graphs of girth at least 5 and chromatic index 4 whose edge-set cannot be covered by four perfect matchings. Only two such graphs were known. It turns out that the family F also has interesting properties with respect to the shortest cycle cover problem. The shortest cycle cover of any cubic bridgeless graph with m edges has length at least , and we show that this inequality is strict for graphs of F. We also construct the first known snark with no cycle cover of length less than .

Three-coloring graphs with no induced seven-vertex path I : the triangle-free case

Abstract: In this paper, we give a polynomial time algorithm which determines if a given triangle-free graph with no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists.

Weighted Coloring in Trees

Abstract: A proper coloring of a graph is a partition of its vertex set into stable sets, where each part corresponds to a color. For a vertex-weighted graph, the weight of a color is the maximum weight of its vertices. The weight of a coloring is the sum of the weights of its colors. Guan and Zhu defined the weighted chromatic number of a vertex-weighted graph $G$ as the smallest weight of a proper coloring of $G$. If vertices of a graph have weight 1, its weighted chromatic number coincides with its chromatic number. Thus, the problem of computing the weighted chromatic number, a.k.a. the max coloring problem, is NP-hard in general graphs. It remains NP-hard in some graph classes as bipartite graphs. Approximation algorithms have been designed in several graph classes; in particular, there exists a polynomial-time approximation scheme for trees. Surprisingly, the time-complexity of computing this parameter in trees is still open. The exponential time hypothesis (ETH) states that 3-SAT cannot be solved in subexpon...