Abstract: The focus of this monograph is on symmetry breaking problems in the message-passing model of distributed computing. In this model a communication network is represented by a n-vertex graph G = (V,E), whose vertices host autonomous processors. The processors communicate over the edges of G in discrete rounds. The goal is to devise algorithms that use as few rounds as possible. A typical symmetry-breaking problem is the problem of graph coloring. Denote by [delta] the maximum degree of G. While coloring G with [delta]+ 1 colors is trivial in the centralized setting, the problem becomes much more challenging in the distributed one. One can also compromise on the number of colors, if this allows for more efficient algorithms. Other typical symmetry-breaking problems are the problems of computing a maximal independent set (MIS) and a maximal matching (MM). The study of these problems dates back to the very early days of distributed computing. The founding fathers of distributed computing laid firm foundations for the area of distributed symmetry breaking already in the eighties. In particular, they showed that all these problems can be solved in randomized logarithmic time. Also, Linial showed that an O([delta]2)-coloring can be solved very efficiently deterministically. However, fundamental questions were left open for decades. In particular, it is not known if the MIS or the ([delta] + 1)-coloring can be solved in deterministic polylogarithmic time. Moreover, until recently it was not known if in deterministic polylogarithmic time one can color a graph with significantly fewer than [delta]2 colors. Additionally, it was open (and still open to some extent) if one can have sublogarithmic randomized algorithms for the symmetry breaking problems. Recently, significant progress was achieved in the study of these questions. More efficient deterministic and randomized ([delta] + 1)-coloring algorithms were achieved. Deterministic [delta]1 + o(1)-coloring algorithms with polylogarithmic running time were devised. Improved (and often sublogarithmic-time) randomized algorithms were devised. Drastically improved lower bounds were given. Wide families of graphs in which these problems are solvable much faster than on general graphs were identified. The objective of our monograph is to cover most of these developments, and as a result to provide a treatise on theoretical foundations of distributed symmetry breaking in the message-passing model. We hope that our monograph will stimulate further progress in this exciting area. Table of Contents: Acknowledgments / Introduction / Basics of Graph Theory / Basic Distributed Graph Coloring Algorithns / Lower Bounds / Forest-Decomposition Algorithms and Applications / Defective Coloring / Arbdefective Coloring / Edge-Coloring and Maximal Matching / Network Decompositions / Introduction to Distributed Randomized Algorithms / Conclusion and Open Questions / Bibliography / Authors' Biographies

Abstract: We present a novel upper bound for the optimal index coding rate. Our bound uses a graph theoretic quantity called the local chromatic number. We show how a good local coloring can be used to create a good index code. The local coloring is used as an alignment guide to assign index coding vectors from a general position MDS code. We further show that a natural LP relaxation yields an even stronger index code. Our bounds provably outperform the state of the art on index coding but at most by a constant factor.

Abstract: Let H = (V, E) be a hypergraph. A conflict-free coloring of H is an assignment of colors to V such that, in each hyperedge e ∈ E, there is at least one uniquely-colored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to cellular antennae, in battery consumption aspects of sensor networks, in RFID protocols, and several other fields. Conflict-free coloring has been the focus of many recent research papers. In this paper, we survey this notion and its combinatorial and algorithmic aspects.

Abstract: If the vertices of a graph G are colored with k colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then G is said to be equitably k-colorable. The equitable chromatic number D .G/ is the smallest integer k such that G is equitably k-colorable. In the first introduction section, results obtained about the equitable chromatic number before 1990 are surveyed. The research on equitable coloring has attracted enough attention only since the early 1990s. In the subsequent sections, positive evidence for the important equitable -coloring conjecture is supplied from graph classes such as forests, split graphs, outerplanar graphs, series-parallel graphs, planar graphs, graphs with low degeneracy, graphs with bounded treewidth, Kneser graphs, and interval graphs. Then three kinds of graph products are investigated. A list version of equitable coloring is introduced. The equitable coloring is further examined in the wider context of graph packing. Appropriate conjectures for equitable -coloring of disconnected graphs are then studied. Variants of the well-known and significant Hajnal and Szemeredi Theorem are discussed. A brief summary of applications of equitable coloring is given. Related notions, such as equitable edge coloring, equitable total coloring, equitable defective coloring, and equitable coloring of uniform hypergraphs, are touched upon. This chapter ends with a short conclusion section. This survey is an updated version of Lih [102].

Abstract: This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing [email protected]?s study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the no-answer to an instance of q-List Coloring on F.

Abstract: Let k be an integer. Two vertex k-colorings of a graph are adjacent if they differ on exactly one vertex. A graph is k-mixing if any proper k-coloring can be transformed into any other through a sequence of adjacent proper k-colorings. Any graph is ( t w + 2 ) -mixing, where tw is the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two ( t w + 2 ) -colorings is at most quadratic, a problem left open in Bonamy et al. (2012). Jerrum proved that any graph is k-mixing if k is at least the maximum degree plus two. We improve Jerrumʼs bound using the grundy number, which is the worst number of colors in a greedy coloring.

Abstract: This note relates to bounds on the chromatic number χ(R) of the Euclidean space, which is the minimum number of colors needed to color all the points in R so that any two points at the distance 1 receive different colors. In [6] a sequence of graphs Gn in R was introduced showing that χ(R) ≥ χ(Gn) ≥ (1 + o(1)) 2 6 . For many years, this bound has been remaining the best known bound for the chromatic numbers of some low-dimensional spaces. Here we prove that χ(Gn) ∼ n 2 6 and find an exact formula for the chromatic number in the case of n = 2 and n = 2 − 1.

Abstract: Graph coloring problem is a classical example for NP-hard combinatorial optimization. Solution to this graph coloring problem often finds its applications to various engineering fields. This paper exhibits the robustness of genetic algorithm to solve a graph coloring. The proposed genetic algorithm employs an innovative single parent conflict gene crossover and a conflict gene mutation as its operators. The time taken to get a convergent solution of this proposed genetic method has been compared with the existing approaches and has been proved to be effective. The performance of this approximation method is evaluated using some benchmarking graphs, and are found to be competent.

Abstract: Let $HP_{n,m,k}$ be drawn uniformly from all $k$-uniform, $k$-partite hypergraphs where each part of the partition is a disjoint copy of $[n]$. We let $HP^{(\k)}_{n,m,k}$ be an edge colored version, where we color each edge randomly from one of $\k$ colors. We show that if $\k=n$ and $m=Kn\log n$ where $K$ is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if $n$ is even and $m=Kn\log n$ where $K$ is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in $G^{(n)}_{n,m}$. Here $G^{(n)}_{n,m}$ denotes a random edge coloring of $G_{n,m}$ with $n$ colors. When $n$ is odd, our proof requires $m=\om(n\log n)$ for there to be a rainbow Hamilton cycle.

Abstract: A star coloring of an undirected graph G is a proper vertex coloring of G i.e., no two adjacent vertices are assigned the same color such that no path on four vertices is 2-colored. The star chromatic number of G is the smallest integer k for which G admits a star coloring with k colors. In this paper, we prove that every subcubic graph is 6-star-colorable. Moreover, the upper bound 6 is best possible, based on the example constructed by Fertin, Raspaud, and Reed J Graph Theory 473 2004, 140-153.

Abstract: In this paper we discuss the game coloring of planar graphs. This parameter provides an upper bound for the game chromatic number of graph. We describe the problem and its solution given by Xuding Zhu (1) and point out an error in it.

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 .

Abstract: An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiamik (Math. Slovaca 28 (1978), 139–145) and later Alon et al. (J Graph Theory 37 (2001), 157–167) conjectured that for any simple graph G with maximum degree Δ. In this article, we confirm this conjecture for planar graphs of girth at least 4.

Abstract: An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a^'(G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. It was conjectured that a^'(G)@[email protected]+2 for any simple graph G with maximum degree @D. In this paper, we prove that if G is a planar graph, then a^'(G)@[email protected]+7. This improves a result by Basavaraju et al. [M. Basavaraju, L.S. Chandran, N. Cohen, F. Havet, T. Muller, Acyclic edge-coloring of planar graphs, SIAM J. Discrete Math. 25 (2011) 463-478], which says that every planar graph G satisfies a^'(G)@[email protected]+12.

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 paper we prove that deciding whether this number is at most 4 for a given cubic bridgeless graph is NP-complete We also construct an infinite family $\cal F$ of snarks (cyclically 4-edge-connected cubic graphs of girth at least five and chromatic index four) whose edge-set cannot be covered by 4 perfect matchings Only two such graphs were known It turns out that the family $\cal F$ 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 $\tfrac43m$, and we show that this inequality is strict for graphs of $\cal F$ We also construct the first known snark with no cycle cover of length less than $\tfrac43m+2$

Abstract: This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with the same birthday?). It is shown that if the number of colors grows to infinity, the asymptotic distribution is either a Poisson mixture or a Normal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors. This result holds for any graph sequence, deterministic or random. On the other hand, when the number of colors is fixed, a necessary and sufficient condition for asymptotic normality is determined. Finally, using some results from the emerging theory of dense graph limits, the asymptotic (non-normal) distribution is characterized for any converging sequence of dense graphs. The proofs are based on moment calculations which relate to the results of Erd\H os and Alon on extremal subgraph counts. As a consequence, a simpler proof of a result of Alon, estimating the number of isomorphic copies of a cycle of given length in graphs with a fixed number of edges, is presented.

Abstract: A vertex coloring of a graph G is an assignment of colors to the vertices of G so that every two adjacent vertices of G have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set S of vertices of a graph G is a dominating set in G if every vertex outside of S is adjacent to at least one vertex belonging to S. A domination parameter of G is related to those structures of a graph that satisfy some domination property together with other conditions on the vertices of G. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-k colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, through some relationships between the distance-k chromatic number of corona graphs and the distance-k chromatic number of its factors. Moreover, we give the exact value of the distance-k chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating–domination number of corona graphs. We give closed formulaes for the k-domination number, the distance-k domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.

Abstract: We model the problem of channel assignment in mobile networks as one of temporal coloring (T-coloring), that is, coloring a time-varying graph. In order to capture the impact of channel re-assignments due to mobility, we model the cost of coloring as C + αA, where C is the total number of colors used and A is the total number of color changes, and α is a user-selectable parameter reflecting the relative penalty of channel usage and re-assignments. Using these models, we present several novel algorithms for temporal coloring. We begin by analyzing two simple algorithms called SNAP and SMASH that take diametrically opposite positions on colors vs re-assignments, and provide theoretical results on the ranges of α in which one outperforms the other, both for arbitrary and random time-varying graphs. We then present six more algorithms that build upon each of SNAP and SMASH in different ways. Simulations on random geometric graphs with random waypoint mobility show that the relative cost of the algorithms depends upon the value of α and the transmission range, and we identify precise values at which the crossovers happen.

Abstract: A graph G is equitably k-choosable if for every k-list assignment L there exists an L-coloring of G such that every color class has at most vertices. We prove results toward the conjecture that every graph with maximum degree at most r is equitably -choosable. In particular, we confirm the conjecture for and show that every graph with maximum degree at most r and at least r3 vertices is equitably -choosable. Our proofs yield polynomial algorithms for corresponding equitable list colorings.

Abstract: A coloring of the edges of a graph $G$ is strong if each color class is an induced matching of $G$. The strong chromatic index of $G$, denoted by $\chi_{s}^{\prime}(G)$, is the least number of colors in a strong edge coloring of $G$. In this note we prove that $\chi_{s}^{\prime}(G)\leq (4k-1)\Delta (G)-k(2k+1)+1$ for every $k$-degenerate graph $G$. This confirms the strong version of conjecture stated recently by Chang and Narayanan [3]. Our approach allows also to improve the upper bound from [3] for chordless graphs. We get that $% \chi_{s}^{\prime}(G)\leq 4\Delta -3$ for any chordless graph $G$. Both bounds remain valid for the list version of the strong edge coloring of these graphs.

Abstract: An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a′(G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiamcik (Math. Slovaca 28:139–145, 1978) and later Alon, Sudakov and Zaks (J. Graph Theory 37:157–167, 2001) conjectured that a′(G)≤Δ+2 for any simple graph G with maximum degree Δ. In this paper, we confirm this conjecture for planar graphs G with Δ≠4 and without 4-cycles.

Abstract: We study the Max k-colored clustering problem, where, given an edge-colored graph with k colors, we seek to color the vertices of the graph so as to find a clustering of the vertices maximizing the number (or the weight) of matched edges, i.e. the edges having the same color as their extremities. We show that the cardinality problem is NP-hard even for edge-colored bipartite graphs with a chromatic degree equal to two and k ≥ 3. Our main result is a constant approximation algorithm for the weighted version of the Max k-colored clustering problem which is based on a rounding of a natural linear programming relaxation. For graphs with chromatic degree equal to two, we improve this ratio by exploiting the relation of our problem with the Max 2-and problem. We also present a reduction to the maximum-weight independent set (IS) problem in bipartite graphs which leads to a polynomial time algorithm for the case of two colors.

Abstract: We extend the ideas of Snevily and Avgustinovitch to enlarge the families of 2mregular graphs and m-regular bipartite graphs that are known to decompose into isomorphic copies of a tree T with m edges. For example, consider r1, . . . , rk with ∑k i=1 ri = m. If T has a k-edge-coloring with ri edges of color i such that every path in T uses some color once or twice, then every cartesian product of graphs G1, . . . , Gk such that Gi is 2ri-regular for 1 ≤ i ≤ k decomposes into copies of T .

Abstract: Let R be a commutative ring. The unit graph of R, denoted by G(R), is a graph with all elements of R as vertices and two distinct vertices x, y ∈ R are adjacent if and only if x + y ∈ U(R) where U(R) denotes the set of all units of R. In this paper, we examine the preservation of the connectedness, diameter, girth, and some other properties, such as chromatic index, clique number and planarity of the unit graph G(R) under extensions to polynomial and power series rings.