Abstract: We give an efficient local search algorithm that computes a good vertex coloring of a graph G. In order to better illustrate this local search method, we view local moves as selfish moves in a suitably defined game. In particular, given a graph G = (V,E) of n vertices and m edges, we define the graph coloring game Γ(G) as a strategic game where the set of players is the set of vertices and the players share the same action set, which is a set of n colors. The payoff that a vertex v receives, given the actions chosen by all vertices, equals the total number of vertices that have chosen the same color as v, unless a neighbor of v has also chosen the same color, in which case the payoff of v is 0. We show: The game Γ(G) has always pure Nash equilibria. Each pure equilibrium is a proper coloring of G. Furthermore, there exists a pure equilibrium that corresponds to an optimum coloring. We give a polynomial time algorithm $\mathcal{A}$ which computes a pure Nash equilibrium of Γ(G). The total number, k, of colors used in any pure Nash equilibrium (and thus achieved by $\mathcal{A}$) is $k\leq\min\{\Delta_2+1, \frac{n+\omega}{2}, \frac{1+\sqrt{1+8m}}{2}, n-\alpha+1\}$, where ω, α are the clique number and the independence number of G and Δ 2 is the maximum degree that a vertex can have subject to the condition that it is adjacent to at least one vertex of equal or greater degree. (Δ 2 is no more than the maximum degree ” of G.) Thus, in fact, we propose here a new, efficient coloring method that achieves a number of colors satisfying (together) the known general upper bounds on the chromatic number X. Our method is also an alternative general way of proving, constructively, all these bounds. Finally, we show how to strengthen our method (staying in polynomial time) so that it avoids "bad" pure Nash equilibria (i.e. those admitting a number of colors k far away from X). In particular, we show that our enhanced method colors optimally dense random q-partite graphs (of fixed q) with high probability.

Abstract: An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge 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 $G$, $a'(G)\le \Delta+2$, where $\Delta =\Delta(G)$ denotes the maximum degree of $G$. We prove the conjecture for connected graphs with $\Delta(G) \le 4$, with the additional restriction that $m \le 2n-1$, where $n$ is the number of vertices and $m$ is the number of edges in $G $. Note that for any graph $G$, $m \le 2n$, when $\Delta(G) \le 4$. It follows that for any graph $G$ if $\Delta(G) \le 4$, then $a'(G) \le 7$.

Abstract: This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G* be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G* has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product GsG′ has game chromatic number at most k(k + m - 1). As a consequence, the Cartesian product of two forests has game chromatic number at most 10, and the Cartesian product of two planar graphs has game chromatic number at most 105. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 261–278, 2008

Abstract: In this paper we present a new channel allocation scheme for IEEE 802.11 based mesh networks with point-to- point links, designed for rural areas. Our channel allocation scheme allows continuous full-duplex data transfer on every link in the network. Moreover, we do not require any synchronization across the links as the channel assignment prevents cross link interference. Our approach is simple. We consider any link in the network as made up of two directed edges. To each directed edge at a node, we assign a non-interfering IEEE 802.11 channel so that the set of channels assigned to the outgoing edges is disjoint from channels assigned to the incoming edges. Evaluation of this scheme in a testbed demonstrate throughput gains of between 50 - 100%, and significantly less end-to-end delays, over existing link scheduling/channel allocation protocols (such as 2P [11]) designed for point-to-point mesh networks. Formally speaking, this channel allocation scheme is equivalent to an edge-coloring problem, that we call the directed edge coloring (DEC) problem. We establish a relationship between this coloring problem and the classical vertex coloring problem, and thus, show that this problem is NP-hard. More precisely, we give an algorithm that, given k vertex coloring of a graph can directed edge color it using xi(k) colors, where xi(k) is the smallest integer n such that (lfloorn/2rfloor/n ) ges k.

Abstract: Let G be a planar graph with maximum degree Δ. It is proved that if Δ ≥ 8 and G is free of k-cycles for some k ∈ {5,6}, then the total chromatic number χ′′(G) of G is Δ + 1.

Abstract: A star coloring of a graph is a proper coloring such that no path on four vertices is 2-colored. We prove that every planar graph with girth at least 9 can be star colored using 5 colors, and that every planar graph with girth at least 14 can be star colored using 4 colors; the figure 4 is best possible. We give an example of a girth 7 planar graph that requires 5 colors to star color.

Abstract: Packing coloring is a partitioning of the vertex set of a graph with the property that vertices in the i -th class have pairwise distance greater than i . We solve an open problem of Goddard et al. and show that the decision whether a tree allows a packing coloring with at most k classes is NP-complete. We accompany this NP-hardness result by a polynomial time algorithm for trees for closely related variant of the packing coloring problem where the lower bounds on the distances between vertices inside color classes are determined by an infinite nondecreasing sequence of bounded integers.

Abstract: Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number χg(G) is the minimum k for which the first player has a winning strategy. In this study, we analyze the asymptotic behavior of this parameter for a random graph Gn,p. We show that with high probability, the game chromatic number of Gn,p is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008

Abstract: We show that the vertices of any plane graph in which every face is of size at least g can be colored by (3g Aý 5)=4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs that admit no vertex coloring of this type with more than (3g+1)=4 colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by 3 colors in which all colors appear in every face is NP-complete even for graphs in which all faces are of size 3 or 4 only. If all faces are of size 3 this can be decided in polynomial time.

Abstract: An embedding of a 3-regular graph in a surface is called polyhedral if its dual is a simple graph. An old graph-coloring conjecture is that every 3-regular graph with a polyhedral embedding in an orientable surface has a 3-edge-coloring. An affirmative solution of this problem would generalize the dual form of the Four Color Theorem to every orientable surface. In this paper we present a negative solution to the conjecture, showing that for each orientable surface of genus at least 5, there exist infinitely many 3-regular non-3-edge-colorable graphs with a polyhedral embedding in the surface.

Abstract: Sierpi\'nski graphs $S(n,3)$ are the graphs of the Tower of Hanoi puzzle with $n$ disks, while Sierpi\'nski gasket graphs $S_n$ are the graphs naturally defined by the finite number of iterations that lead to the Sierpi\'nski gasket. An explicit labeling of the vertices of $S_n$ is introduced. It is proved that $S_n$ is uniquely 3-colorable, that $S(n,3)$ is uniquely 3-edge-colorable, and that $\chi'(S_n)=4$, thus answering a question from~[15]. It is also shown that $S_n$ contains a 1-perfect code only for $n=1$ or $n=3$ and that every $S(n,3)$ contains a unique Hamiltonian cycle.

Abstract: We consider graph coloring problems where the cost of a coloring is the sum of the costs of the colors, and the cost of a color is a monotone concave function of the total weight of the class. This models resource allocation problems where the cost of a resource depends on the use of the resource. The specific case of interval graphs is of special interest as multi-criteria interval scheduling. We give an algorithm for all perfect graphs that yields a robust coloring: a particular solution that simultaneously approximates all concave functions. For graphs with uniform weights, we show how to modify the solution to approximate any monotone cost function. We complement these results with a number of hardness results and some exact algorithms on restricted classes of graphs.

Abstract: A plane graph is l-facially $k$-colorable if its vertices can be colored with $k$ colors such that any two distinct vertices on a facial segment of length at most lare colored differently. We prove that every plane graph is $3$-facially $11$-colorable. As a consequence, we derive that every $2$-connected plane graph with maximum face-size at most $7$ is cyclically $11$-colorable. These two bounds are just one higher than those that are proposed by the $(3\l+1)$-conjecture and the cyclic conjecture.

Abstract: We are interested in coloring the vertices of a mixed graph, i.e., a graph containing edges and arcs. We consider two different coloring problems: in the first one, we want adjacent vertices to have different colors and the tail of an arc to get a color strictly less than a color of the head of this arc; in the second problem, we also allow vertices linked by an arc to have the same color. For both cases, we present bounds on the mixed chromatic number and we give some complexity results which strengthen earlier results given in [B. Ries, Coloring some classes of mixed graphs, Discrete Applied Mathematics 155 (2007) 1-6].

Abstract: Some sufficient conditions (in terms of the girth and maximum degree) are given for the list 2-distance chromatic number of a planar graph with maximum degree Δ to be equal to Δ + 1.

Abstract: We study an information-theoretic variant of the graph coloring problem in which the objective function to minimize is the entropy of the coloring. The minimum entropy of a coloring is called the chromatic entropy and was shown by Alon and Orlitsky (IEEE Trans. Inform. Theory 42(5):1329–1339, 1996) to play a fundamental role in the problem of coding with side information. In this paper, we consider the minimum entropy coloring problem from a computational point of view. We first prove that this problem is NP-hard on interval graphs. We then show that, for every constant e>0, it is NP-hard to find a coloring whose entropy is within (1−e)log n of the chromatic entropy, where n is the number of vertices of the graph. A simple polynomial case is also identified. It is known that graph entropy is a lower bound for the chromatic entropy. We prove that this bound can be arbitrarily bad, even for chordal graphs. Finally, we consider the minimum number of colors required to achieve minimum entropy and prove a Brooks-type theorem.

Abstract: A batch is a set of jobs that start execution at the same time; only when the last job is completed can the next batch be started. When there are constraints or conflicts between the jobs, we need to ensure that jobs in the same batch be non-conflicting. That is, we seek a coloring of the conflict graph. The two most common objectives of schedules and colorings are the makespan, or the maximum job completion time, and the sum of job completion times. This gives rise to two types of batch coloring problems: max-coloringand batch sum coloring, respectively. We give the first polynomial time approximation schemesfor batch sum coloring on several classes of "non-thick" graphs that arise in applications. This includes paths, trees, partial k-trees, and planar graphs. Also, we give an improved O(nlogn) exact algorithm for the max-coloring problem on paths.

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