Abstract: A synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with constant outdegree of all its vertices if the greatest common divisor of lengths of all its cycles is one. The problem was posed by Adler, Goodwyn and Weiss over 30 years ago and evoked noticeable interest among the specialists in the theory of graphs, deterministic automata and symbolic dynamics. The positive solution of the road coloring problem is presented.

Abstract: Distributed greedy coloring is an interesting and intuitive variation of the standard coloring problem. Given an order among the colors, a coloring is said to be greedy if there does not exist a vertex for which its associated color can be replaced by a color of lower position in the fixed order without violating the property that neighboring vertices must receive different colors. We consider the problems of Greedy Coloring and Largest First Coloring (a variant of greedy coloring with strengthened constraints) in the Linial model of distributed computation, providing lower and upper bounds and a comparison to the (Δ + 1)-Coloring and Maximal Independent Set problems, with Δ being the maximum vertex degree in G. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009 Some of the results of this paper were announced in [11], [16].

Abstract: Given an edge coloring F of a graph G, a vertex coloring of G is adapted to F if no color appears at the same time on an edge and on its two endpoints. If for some integer k, a graph G is such that given any list assignment L to the vertices of G, with |L(v)|gk for all v, and any edge coloring F of G, G admits a coloring c adapted to F where c(v)∈L(v) for all v, then G is said to be adaptably k-choosable. In this note, we prove that K5-minor-free graphs are adaptably 4-choosable, which implies that planar graphs are adaptably 4-colorable and answers a question of Hell and Zhu. We also prove that triangle-free planar graphs are adaptably 3-choosable and give negative results on planar graphs without 4-cycle, planar graphs without 5-cycle, and planar graphs without triangles at distance t, for any tg0. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 127–138, 2009

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 l2, 4, 6, 8r. We provide several sufficient conditions for the existence of such a subgraph. © 2009 Wiley Periodicals, Inc. J Graph Theory

Abstract: It was only recently shown by Shi and Wormald, using the differential equation method to analyze an appropriate algorithm, that a random 5-regular graph asymptotically almost surely has chromatic number at most 4. Here, we show that the chromatic number of a random 5-regular graph is asymptotically almost surely equal to 3, provided a certain four-variable function has a unique maximum at a given point in a bounded domain. We also describe extensive numerical evidence that strongly suggests that the latter condition holds. The proof applies the small subgraph conditioning method to the number of locally rainbow balanced 3-colorings, where a coloring is balanced if the number of vertices of each color is equal, and locally rainbow if every vertex is adjacent to at least one vertex of each of the other colors. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 157–191, 2009

Abstract: A b-coloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The b-chromatic number of a graph G, denoted by χ b (G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is b-continuous if it admits a b-coloring with t colors, for every $$t = \chi(G), \ldots, \chi_b(G)$$. We define a graph G to be b-monotonic if χ b (H 1) ≥ χb (H 2) for every induced subgraph H 1 of G, and every induced subgraph H 2 of H 1. In this work, we prove that P 4-sparse graphs (and, in particular, cographs) are b-continuous and b-monotonic. Besides, we describe a dynamic programming algorithm to compute the b-chromatic number in polynomial time within these graph classes.

Abstract: A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G Alon et al conjectured that a′(G) ⩽ Δ(G) + 2 for any graphs For planar graphs G with girth g(G), we prove that a′(G) ⩽ max{2Δ(G) − 2, Δ(G) + 22} if g(G) ⩾ 3, a′(G) ⩽ Δ(G) + 2 if g(G) ⩾ 5, a′(G) ⩽ Δ(G) + 1 if g(G) ⩾ 7, and a′(G) = Δ(G) if g(G) ⩾ 16 and Δ(G) ⩾ 3 For series-parallel graphs G, we have a′(G) ⩽ Δ(G) + 1

Abstract: A face of an edge-colored plane graph is called rainbow if the number of colors used on its edges is equal to its size. The maximum number of colors used in an edge coloring of a connected plane graph Gwith no rainbow face is called the edge-rainbowness of G. In this paper we prove that the edge-rainbowness of Gequals the maximum number of edges of a connected bridge face factor H of G, where a bridge face factor H of a plane graph Gis a spanning subgraph H of Gin which every face is incident with a bridge and the interior of any one face f∈F(G) is a subset of the interior of some face f′∈F(H). We also show upper and lower bounds on the edge-rainbowness of graphs based on edge connectivity, girth of the dual graphs, and other basic graph invariants. Moreover, we present infinite classes of graphs where these equalities are attained. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 84–99, 2009

Abstract: 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 that for any simple and finite graph G, a′(G)lΔ + 2, where Δ=Δ(G) denotes the maximum degree of G. We prove the conjecture for connected graphs with Δ(G)l4, with the additional restriction that ml2n-1, where n is the number of vertices and m is the number of edges in G. Note that for any graph G, ml2n, when Δ(G)l4. It follows that for any graph G if Δ(G)l4, then a′(G)l7. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 192–209, 2009

Abstract: Graph and hypergraph colorings constitute an important subject in combinatorics and algorithm theory. In this work, we study conflict-free coloring for hypergraphs. Conflict-free coloring is one possible generalization of traditional graph coloring. Conflict-free coloring hypergraphs induced by geometric shapes, like intervals on the line, or disks on the plane, has applications in frequency assignment in cellular networks. Colors model frequencies and since the frequency spectrum is limited and expensive, the goal of an algorithm is to minimize the number of assigned frequencies, that is, reuse frequencies as much as possible. We concentrate on an online variation of the problem, especially in the case where the hypergraph is induced by intervals. For deterministic algorithms, we introduce a hierarchy of models ranging from static to online and we compute lower and upper bounds on the numbers of colors used. In the randomized oblivious adversary model, we introduce a framework for conflict-free coloring a specific class of hypergraphs with a logarithmic number of colors. This specific class includes many hypergraphs arising in geometry and gives online randomized algorithm that use fewer colors and fewer random bits than other algorithms in the literature. Based on the same framework, we initiate the study of online deterministic algorithms that recolor few points. For the problem of conlict-free coloring points with respect to a given set of intervals, we describe an efficient algorithm that computes a coloring with at most twice the number of colors of an optimal coloring. We also show that there is a family of inputs that force our algorithm to use two times the number of colors of an optimal solution. Then, we study conflict-free coloring problems in graphs. We compare conflict-free coloring with respect to paths of graphs to a closely related problem, called vertex ranking, or ordered coloring. For conflict-free coloring with respect to neighborhoods of vertices of graphs, we prove that number of colors in the order of the square root of the number of vertices is sufficient and sometimes necessary. Finally, we initiate the study of Ramsey-type problems for conflict-free colorings and compute a van der Waerden-like number.

Abstract: An $r$-edge-coloring of a graph is an assignment of $r$ colors to the edges of the graph An exactly $r$-edge-coloring of a graph is an $r$-edge-coloring of the graph that uses all $r$ colors A matching of an edge-colored graph is called rainbow matching , if no two edges have the same color in the matching In this paper, we prove that an exactly $r$-edge-colored complete graph of order $n$ has a rainbow matching of size $k(\ge 2)$ if $r \ge max\{{2k-3\choose 2}+2, {k-2\choose 2}+(k-2)(n-k+2)+2 \}$, $k \ge 2$, and $n \ge 2k+1$ The bound on $r$ is best possible

Abstract: We present randomized algorithms for online conflict-free coloring (CF in short) of points in the plane, with respect to halfplanes, congruent disks, and nearly-equal axis-parallel rectangles. In all three cases, the coloring algorithms use O(log n) colors, with high probability.We also present a deterministic algorithm for online CF coloring of points in the plane with respect to nearly-equal axis-parallel rectangles, using O(log3n) colors. This is the first efficient (i.e, using polylog(n) colors) deterministic online CF coloring algorithm for this problem.

Abstract: 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 χk(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 χk 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 χk for unit interval graphs. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009

Abstract: Many classes of graphs where the vertex coloring problem is polynomially solvable are known, the most prominent being the class of perfect graphs. However, the list-coloring problem is NP-complete for many subclasses of perfect graphs. In this work we explore the complexity boundary between vertex coloring and list-coloring on such subclasses of perfect graphs where the former admits polynomial-time algorithms but the latter is NP-complete. Our goal is to analyze the computational complexity of coloring problems lying “between” (from a computational complexity viewpoint) these two problems: precoloring extension, μ-coloring, and (γ,μ)-coloring.

Abstract: We consider the following question of Bollobas: given an r-coloring of E(Kn), how large a k-connected subgraph can we find using at most s colors? We provide a partial solution to this problem when s=1 (and n is not too small), showing that when r=2 the answer is n-2k+2, when r=3 the answer is ⌊(n-k)-2⌋+1 or ⌈(n-k)-2⌉+1, and when r-1 is a prime power then the answer lies between n-(r-1)-11(k2-k)r and (n-k+1)-(r-1)+r. The case s≥2 is considered in a subsequent paper (Liu et al.[6]), where we also discuss some of the more glaring open problems relating to this question. © 2009 Wiley Periodicals, Inc. J. Graph Theory 61: 22-44, 2009 The work reported in this paper was done when the authors were at the University of Memphis.

Abstract: 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 bipartite planar graph can be star colored from lists of size 14, and we give an example of a bipartite planar graph that requires at least eight colors to star color. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 1–10, 2009

Abstract: In this paper we study the connectivity problem for wireless networks under the Signal to Interference plus Noise Ratio (SINR) model. Given a set of radio transmitters distributed in some area, we seek to build a directed strongly connected communication graph, and compute an edge coloring of this graph such that the transmitter-receiver pairs in each color class can communicate simultaneously. Depending on the interference model, more or less colors, corresponding to the number of frequencies or time slots, are necessary. We consider the SINR model that compares the received power of a signal at a receiver to the sum of the strength of other signals plus ambient noise . The strength of a signal is assumed to fade polynomially with the distance from the sender, depending on the so-called path-loss exponent $\alpha$. We show that, when all transmitters use the same power, the number of colors needed is constant in one-dimensional grids if $\alpha>1$ as well as in two-dimensional grids if $\alpha>2$. For smaller path-loss exponents and two-dimensional grids we prove upper and lower bounds in the order of $\mathcal{O}(\log n)$ and $\Omega(\log n/\log\log n)$ for $\alpha=2$ and $\Theta(n^{2/\alpha-1})$ for $\alpha<2$ respectively. If nodes are distributed uniformly at random on the interval $[0,1]$, a \emph{regular} coloring of $\mathcal{O}(\log n)$ colors guarantees connectivity, while $\Omega(\log \log n)$ colors are required for any coloring.

Abstract: Grotzsch's theorem states that every triangle-free planar graph is 3-colorable, and several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to find a 3-coloring, but it is not clear how to find such a coloring in linear time (Kowalik used a nontrivial data structure to construct an O(n log n) algorithm).We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement. As a by-product we give a yet simpler proof of Grotzsch's theorem.

Abstract: A set coloring of the graph G is an assignment (function) of distinct subsets of a finite set X of colors to the vertices of the graph, where the colors of the edges are obtained as the symmetric differences of the sets assigned to their end vertices which are also distinct. A set coloring is called a strong set coloring if sets on the vertices and edges are distinct and together form the set of all nonempty subsets of X. A set coloring is called a proper set coloring if all the nonempty subsets of X are obtained on the edges. A graph is called strongly set colorable (properly set colorable) if it admits a strong set coloring (proper set coloring). In this paper we give some necessary conditions for a graph to admit a strong set coloring (a proper set coloring), characterize strongly set colorable complete bipartite graphs and strongly (properly) set colorable complete graphs, etc. Also, we give a construction of a planar strongly set colorable graph from a planar graph, a strongly set colorable tree from a tree and a properly set colorable tree from a tree, etc., thereby showing their embeddings.

Abstract: We present a new method to break symmetry in graph coloring problems. While most alternative techniques add symmetry breaking predicates in a pre-processing step, we developed a learning scheme that translates each encountered conflict into one conflict clause which covers equivalent conflicts arising from any permutation of the colors. Our technique introduces new Boolean variables during the search. For many problems the size of the resolution refutation can be significantly reduced by this technique. Although this is shown for various hand-made refutations, it is rarely used in practice, because it is hard to determine which variables to introduce defining useful predicates. In case of graph coloring, the reason for each conflicting coloring can be expressed as a node in the Zykov-tree, that stems from merging some vertices and adding some edges. So, we focus on variables that represent the Boolean expression that two vertices can be merged (if set to true), or that an edge can be placed between them (if set to false). Further, our algorithm reduces the number of introduced variables by reusing them. We implemented our technique in the state-of-the-art solver minisat. It is competitive with alternative SAT based techniques for graph coloring problems. Moreover, our technique can be used on top of other symmetry breaking techniques. In fact, combined with adding symmetry breaking predicates, huge performance gains are realized.

Abstract: A dynamic coloring of a graph G is a proper coloring such that for every vertex v ∈ V (G) of degree at least 2, the neighbors of v receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number of graphs. In this regard, we shall show that there is a constant c such that for every k-regular graph G, χd(G) ≤ χ(G) + cln k + 1. Also, we introduce an upper bound for the dynamic list chromatic number of regular graphs.

Abstract: Consider the following one-player game. Starting with the empty graph on $n$ vertices, in every step $r$ new edges are drawn uniformly at random and inserted into the current graph. These edges have to be colored immediately with $r$ available colors, subject to the restriction that each color is used for exactly one of these edges. The player's goal is to avoid creating a monochromatic copy of some fixed graph $F$ for as long as possible. We prove explicit threshold functions for the duration of this game for an arbitrary number of colors $r$ and a large class of graphs $F$. This extends earlier work for the case $r=2$ by Marciniszyn, Mitsche, and Stojakovic. We also prove a similar threshold result for the vertex-coloring analogue of this game.

Abstract: We describe a new algorithm, the (k, l)-pebble game with colors, and use it to obtain a characterization of the family of (k, l)-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k, l)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9].

Abstract: We propose a polynomial time approximation algorithm for a novel maximum edge coloring problem which arises from wireless mesh networks [Ashish Raniwala, Tzi-cker Chiueh, Architecture and algorithms for an IEEE 802.11-based multi-channel wireless mesh network, in: INFOCOM 2005, pp. 2223-2234; Ashish Raniwala, Kartik Gopalan, Tzi-cker Chiueh, Centralized channel assignment and routing algorithms for multi-channel wireless mesh networks, Mobile Comput. Commun. Rev. 8 (2) (2004) 50-65]. The problem is to color all the edges in a graph with maximum number of colors under the following q-Constraint: for every vertex in the graph, all the edges incident to it are colored with no more than q ([email protected]?Z,q>=2) colors. We show that the algorithm is a 2-approximation for the case q=2 and a (1+4q-23q^2-5q+2)-approximation for the case q>2 respectively. The case q=2 is of great importance in practice. For complete graphs and trees, polynomial time accurate algorithms are found for them when q=2. The approximation algorithm gives a feasible solution to channel assignment in multi-channel wireless mesh networks.

Abstract: We develop an idea of a local 3-edge-coloring of a cubic graph, a generalization of the usual 3-edge-coloring. We allow for an unlimited number of colors but require that the colors of two edges meeting at a vertex always determine the same third color. Local 3-edge-colorings are described in terms of colorings by points of a partial Steiner triple system such that the colors meeting at each vertex form a triple of the system. An important place in our investigation is held by the two smallest non-trivial Steiner triple systems, the Fano plane PG(2,2) and the affine plane AG(2,3). For i=4,5, and 6 we identify certain configurations F"i and A"i of i lines of the Fano plane and the affine plane, respectively, and prove a theorem saying that a cubic graph admits an F"i-coloring if and only if it admits an A"i-coloring. Among consequences of this is the result of Holroyd and Skoviera [F. Holroyd, M. Skoviera, Colouring of cubic graphs by Steiner triple systems, J. Combin. Theory Ser. B 91 (2004) 57-66] that the edges of every bridgeless cubic graph can be colored by using points and blocks of any non-trivial Steiner triple system S. Another consequence is that every bridgeless cubic graph has a proper edge-coloring by elements of any abelian group of order at least 12 such that around each vertex the group elements sum to 0. We also propose several conjectures concerning edge-coloring of cubic graphs and relate them to several well-known conjectures. In particular, we show that both the Cycle Double Cover Conjecture and the Fulkerson Conjecture can be formulated as a coloring problem in terms of known geometric configurations - the Desargues configuration and the Cremona-Richmond configuration, respectively.