Abstract: The Lovasz local lemma (LLL), introduced by Erdźs 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 works of Alon (Random Struct Algorithms 2(4):367---378, 1991) and Beck (Random Struct Algorithms 2(4):343---365, 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 (J ACM 57(2):11, 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 $$O(\log ^2 n)$$O(log2n) 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 $$epd^2 < 1$$epd2<1 we give a truly simple LLL algorithm running in $$O(\log _{1/epd^2} n)$$O(log1/epd2n) rounds. Under the weaker condition $$ep(d+1) < 1$$ep(d+1)<1, we give a slightly slower algorithm running in $$O(\log ^2 d\cdot \log _{1/ep(d+1)} n)$$O(log2d·log1/ep(d+1)n) rounds. Furthermore, we give an algorithm that runs in sublogarithmic rounds under the condition $$p\cdot f(d) < 1$$p·f(d)<1, where f(d) is an exponential function of d. Although the conditions of the LLL are locally verifiable, we prove that any distributed LLL algorithm requires $${\varOmega }(\log ^* n)$$Ω(logźn) rounds. In many graph coloring problems the existence of a valid coloring is established by one or more applications of the LLL. Using our LLL algorithms, we give logarithmic-time distributed algorithms for frugal coloring, defective coloring, coloring girth-4 (triangle-free) and girth-5 graphs, edge coloring, and list coloring.

Abstract: The complexity of distributed edge coloring depends heavily on the palette size as a function of the maximum degree $\Delta$. In this paper we explore the complexity of edge coloring in the LOCAL model in different palette size regimes. 1. We simplify the \emph{round elimination} technique of Brandt et al. and prove that $(2\Delta-2)$-edge coloring requires $\Omega(\log_\Delta \log n)$ time w.h.p. and $\Omega(\log_\Delta n)$ time deterministically, even on trees. The simplified technique is based on two ideas: the notion of an irregular running time and some general observations that transform weak lower bounds into stronger ones. 2. We give a randomized edge coloring algorithm that can use palette sizes as small as $\Delta + \tilde{O}(\sqrt{\Delta})$, which is a natural barrier for randomized approaches. The running time of the algorithm is at most $O(\log\Delta \cdot T_{LLL})$, where $T_{LLL}$ is the complexity of a permissive version of the constructive Lovasz local lemma. 3. We develop a new distributed Lovasz local lemma algorithm for tree-structured dependency graphs, which leads to a $(1+\epsilon)\Delta$-edge coloring algorithm for trees running in $O(\log\log n)$ time. This algorithm arises from two new results: a deterministic $O(\log n)$-time LLL algorithm for tree-structured instances, and a randomized $O(\log\log n)$-time graph shattering method for breaking the dependency graph into independent $O(\log n)$-size LLL instances. 4. A natural approach to computing $(\Delta+1)$-edge colorings (Vizing's theorem) is to extend partial colorings by iteratively re-coloring parts of the graph. We prove that this approach may be viable, but in the worst case requires recoloring subgraphs of diameter $\Omega(\Delta\log n)$. This stands in contrast to distributed algorithms for Brooks' theorem, which exploit the existence of $O(\log_\Delta n)$-length augmenting paths.

Abstract: We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. We first present a new combinatorial algorithm using O(n4/11) colors. This is the first combinatorial improvement since Blum’s O(n3/8) bound from FOCS’90. Like Blum’s algorithm, our new algorithm composes immediately with recent semi-definite programming approaches, and improves the best bound for the polynomial time algorithm for the coloring of 3-colorable graphs from O(n0.2072) colors by Chlamtac from FOCS’07 to O(n0.2049) colors. Next, we develop a new recursion tailored for combination with semi-definite approaches, bringing us further down to O(n0.19996) colors.

Abstract: The graph coloring problem (GCP) is one of the most studied NP hard problems and has numerous applications. Despite the practical importance of GCP, there are limited works in solving GCP for very large graphs. This paper explores techniques for solving GCP on very large real world graphs.We first propose a reduction rule for GCP, which is based on a novel concept called degree bounded independent set.The rule is iteratively executed by interleaving between lower bound computation and graph reduction. Based on this rule, we develop a novel method called FastColor, which also exploits fast clique and coloring heuristics. We carry out experiments to compare our method FastColor with two best algorithms for coloring large graphs we could find. Experiments on a broad range of real world large graphs show the superiority of our method. Additionally, our method maintains both upper bound and lower bound on the optimal solution, and thus it proves an optimal solution when the upper bound meets the lower bound. In our experiments, it proves the optimal solution for 97 out of 142 instances.

Abstract: We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for $(\Delta+1)$- vertex coloring and $(2\Delta-1)$-edge coloring in a graph with maximum degree $\Delta$. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following three results. (1) We present a randomized algorithm which maintains a $(\Delta+1)$-vertex coloring with $O(\log \Delta)$ expected amortized update time. (2) We present a deterministic algorithm which maintains a $(1+o(1))\Delta$-vertex coloring with $O(\text{poly} \log \Delta)$ amortized update time. (3) We present a simple, deterministic algorithm which maintains a $(2\Delta-1)$-edge coloring with $O(\log \Delta)$ worst-case update time. This improves the recent $O(\Delta)$-edge coloring algorithm with $\tilde{O}(\sqrt{\Delta})$ worst-case update time by Barenboim and Maimon.

Abstract: Let c be a proper edge coloring of a graph G=V,E with integers 1,2,',k. Then ki¾źΔG, while Vizing's theorem guarantees that we can take ki¾źΔG+1. On the course of investigating irregularities in graphs, it has been conjectured that with only slightly larger k, that is, k=ΔG+2, we could enforce an additional strong feature of c, namely that it attributes distinct sums of incident colors to adjacent vertices in G if only this graph has no isolated edges and is not isomorphic to C5. We prove the conjecture is valid for planar graphs of sufficiently large maximum degree. In fact an even stronger statement holds, as the necessary number of colors stemming from the result of Vizing is proved to be sufficient for this family of graphs. Specifically, our main result states that every planar graph G of maximum degree at least 28, which contains no isolated edges admits a proper edge coloring c:Ei¾ź{1,2,',ΔG+1} such that i¾źei¾źucei¾źi¾źei¾źvce for every edge uv of G.

Abstract: The distributed averaging problem is a consensus problem whose objective is to devise a protocol which will enable all the members of a group of autonomous agents to compute the average of the initial values of their individual consensus variables in a distributed manner. Periodic gossiping is a deterministic method for solving the distributed averaging problem by stipulating that each pair of agents which are allowed to gossip, do so repeatedly in accordance with a prespecified periodic schedule. Agent pairs which are allowed to gossip correspond to edges on a given connected, undirected graph. In general, the rate at which the agents' consensus variables converge to the desired average value depends on the order in which the gossips occur over a period. The main contributions of this paper are first to characterize the classes of periodic gossip sequences which have the same convergence rate and second to prove that if the graph of allowable gossips is a tree with each edge restricted to gossiping once per period, the convergence rate is quite surprisingly, fixed and invariant over all possible periodic gossip sequences the graph allows. To arrive at these results, a new and unusual graph theoretic concept, namely the transfer function of a node of an undirected graph, is used. Among all the trees with the same number of edges, optimal tree structures, which yield the fastest convergence rate, can then be sought.

Abstract: We study dynamic graphs in the fully-dynamic centralized setting. In this setting the vertex set of size n of a graph G is fixed, and the edge set changes step-by-step, such that each step either adds or removes an edge. The goal in this setting is maintaining a solution to a certain problem (e.g., maximal matching, edge coloring) after each step, such that each step is executed efficiently. The running time of a step is called update-time. One can think of this setting as a dynamic network that is monitored by a central processor that is responsible for maintaining the solution. Currently, for several central problems, the best-known deterministic algorithms for general graphs are the naive ones which have update-time O(n). This is the case for maximal matching and proper O(∆)-edge-coloring. The question of existence of sublinear in n update-time deterministic algorithms for dense graphs and general graphs remained wide open. We address this question by devising sublinear update-time deterministic algorithms for maximal matching in graphs with bounded neighborhood independence o(n/ log2 n), and for proper O(∆)-edge-coloring in general graphs. The family of bounded neighborhood independence is a very wide family of dense graphs that represent very well various networks. For graphs with constant neighborhood independence, our maximal matching algorithm has O(√n) update-time. Our O(∆)-edge-coloring algorithms has O(√∆) update-time for general graphs. In order to obtain our results we employ a novel approach that adapts certain distributed algorithms of the LOCAL setting to the centralized fully-dynamic setting. This is achieved by optimizing the work each processors performs, and efficiently simulating a distributed algorithm in a centralized setting. The simulation is efficient thanks to a careful selection of the network parts that the algorithm is invoked on, and by deducing the solution from the additional information that is present in the centralized setting, but not in the distributed one. Our experiments on various network topologies and scenarios demonstrate that our algorithms are highly-efficient in practice. We believe that our approach is of independent interest and may be applicable to additional problems.

Abstract: Inspired by a 1987 result of Hanson and Toft [Edge-colored saturated graphs, J Graph Theory 11 (1987), 191–196] and several recent results, we consider the following saturation problem for edge-colored graphs. An edge-coloring of a graph F is rainbow if every edge of F receives a different color. Let R(F) denote the set of rainbow-colored copies of F. A t-edge-colored graph G is (R(F),t)-saturated if G does not contain a rainbow copy of F but for any edge e∈E(G¯) and any color i∈[t], the addition of e to G in color i creates a rainbow copy of F. Let sat t(n,R(F)) denote the minimum number of edges in an (R(F),t)-saturated graph of order n. We call this the rainbow saturation number of F. In this article, we prove several results about rainbow saturation numbers of graphs. In stark contrast with the related problem for monochromatic subgraphs, wherein the saturation is always linear in n, we prove that rainbow saturation numbers have a variety of different orders of growth. For instance, the rainbow saturation number of the complete graph Kn lies between nlogn/loglogn and nlogn, the rainbow saturation number of an n-vertex star is quadratic in n, and the rainbow saturation number of any tree that is not a star is at most linear.

Abstract: An edge colored graph is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number, rc-number for short, of a graph $${\varGamma }$$Γ, is the smallest number of colors that are needed in order to make $${\varGamma }$$Γ rainbow connected. In this paper, we give a method to bound the rc-numbers of graphs with certain structural properties. Using this method, we investigate the rc-numbers of Cayley graphs, especially, those defined on abelian groups and on dihedral groups.

Abstract: Graph coloring is a classical NP-hard combinatorial optimization problem with many practical applications. A broad range of heuristic methods exist for tackling the graph coloring problem: from fast greedy algorithms to more time-consuming metaheuristics. Although the latter produce better results in terms of minimizing the number of colors, the former are widely employed due to their simplicity. These heuristic methods are centralized since they operate by using complete knowledge of the graph. However, in real-world environmets where each component only interacts with a limited number of other components, the only option is to apply decentralized methods. This paper explores a novel and simple algorithm for decentralized graph coloring that uses a fixed number of colors and iteratively reduces the edge conflicts in the graph. We experimentally demonstrate that, for most of the tested instances, the new algorithm outperforms a recent and very competitive algorithm for decentralized graph coloring in terms of coloring quality. In our experiments, the fixed number of colors used by the new algorithm is controlled in a centralized manner.

Abstract: The anti-Ramsey number AR(G, H) is defined to be the maximum number of colors in an edge coloring of G which doesn't contain any rainbow subgraphs isomorphic to H. It is clear that there is an $$AR(K_{m,n},kK_2)$$AR(Km,n,kK2)-edge-coloring of $$K_{m,n}$$Km,n that doesn't contain any rainbow $$kK_2$$kK2. In this paper, we show the uniqueness of this kind of $$AR(K_{m,n},kK_2)$$AR(Km,n,kK2)-edge-coloring of $$K_{m,n}$$Km,n.

Abstract: In the maximum traveling salesman problem (Max TSP) we are given a complete undirected graph with nonnegative weights on the edges and we wish to compute a traveling salesman tour of maximum weight. We present a fast combinatorial \(\frac{4}{5}\) – approximation algorithm for Max TSP. The previous best approximation for this problem was \(\frac{7}{9}\). The new algorithm is based on a technique of eliminating difficult subgraphs via gadgets with half-edges, a new method of edge coloring and a technique of exchanging edges.

Abstract: The degree splitting problem requires coloring the edges of a graph red or blue such that each node has almost the same number of edges in each color, up to a small additive discrepancy. The directed variant of the problem requires orienting the edges such that each node has almost the same number of incoming and outgoing edges, again up to a small additive discrepancy. We present deterministic distributed algorithms for both variants, which improve on their counterparts presented by Ghaffari and Su [SODA'17]: our algorithms are significantly simpler and faster, and have a much smaller discrepancy. This also leads to a faster and simpler deterministic algorithm for $(2+o(1))\Delta$-edge-coloring, improving on that of Ghaffari and Su.

Abstract: Let 𝒢 be a family of graphs. The anti-Ramsey number AR(G,𝒢) for 𝒢 in the graph G is the maximum number of colors in an edge coloring of G that does not have any rainbow copy of any graph in 𝒢. In this paper, we consider the anti-Ramsey number for matchings in regular bipartite graphs and determine its value under several conditions.

Abstract: In the distributed message-passing setting a communication network is represented by a graph whose vertices represent processors that perform local computations and communicate over the edges of the graph In the distributed edge-coloring problem the processors are required to assign colors to edges, such that all edges incident on the same vertex are assigned distinct colors The previously-known deterministic algorithms for edge-coloring employed at least (2Δ - 1) colors, even though any graph admits an edge-coloring with Δ + 1 colors [36] Moreover, the previously-known deterministic algorithms that employed at most O(Δ) colors required superlogarithmic time [3,6,7,17] In the current paper we devise deterministic edge-coloring algorithms that employ only Δ + o(Δ) colors, for a very wide family of graphs Specifically, as long as the arboricity a of the graph is a = O(Δ1 - e), for a constant e > 0, our algorithm computes such a coloring within polylogarithmic deterministic time We also devise significantly improved deterministic edge-coloring algorithms for general graphs for a very wide range of parameters Specifically, for any value κ in the range [4Δ, 2o(log Δ) ⋅ Δ], our κ-edge-coloring algorithm has smaller running time than the best previously-known κ-edge-coloring algorithms Our algorithms are actually much more general, since edge-coloring is equivalent to vertex-coloring of line graphs Our method is applicable to vertex-coloring of the family of graphs with bounded diversity that contains line graphs, line graphs of hypergraphs, and many other graphs We significantly improve upon previous vertex-coloring of such graphs, and as an implication also obtain the improved edge-coloring algorithms for general graphs Our results are obtained using a novel technique that connects vertices or edges in a certain way that reduces clique size The resulting structures, which we call connectors, can be colored more efficiently than the original graph Moreover, the color classes constitute simpler subgraphs that can be colored even more efficiently using appropriate connectors We introduce several types of connectors that are useful for various scenarios We believe that this technique is of independent interest

Abstract: The star chromatic index of a multigraph $G$, denoted $\chi'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $\chi'_{s}(G)\le k$. Dvořak, Mohar and Samal [Star chromatic index, J. Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable. They conjectured in the same paper that every subcubic multigraph should be star $6$-edge-colorable. In this paper, we first prove that it is NP-complete to determine whether $\chi'_s(G)\le3$ for an arbitrary graph $G$. This answers a question of Mohar. We then establish some structure results on subcubic multigraphs $G$ with $\delta(G)\le2$ such that $\chi'_s(G)>k$ but $\chi'_s(G-v)\le k$ for any $v\in V(G)$, where $k\in\{5,6\}$. We finally apply the structure results, along with a simple discharging method, to prove that every subcubic multigraph $G$ is star $6$-edge-colorable if $mad(G)<5/2$, and star $5$-edge-colorable if $mad(G)<24/11$, respectively, where $mad(G)$ is the maximum average degree of a multigraph $G$. This partially confirms the conjecture of Dvořak, Mohar and Samal.

Abstract: In the study of computer science, optimization, computation of Hessians matrix, graph coloring is an important tool. In this paper, we consider a classical coloring, total coloring. Let $$G=(V,E)$$G=(V,E) be a graph. Total coloring is a coloring of $$V\cup {E}$$VźE such that no two adjacent or incident elements (vertex/edge) receive the same color. Let G be a planar graph with $$\varDelta \ge 8$$Δź8. We proved that if for every vertex $$v\in V$$vźV, there exists two integers $$i_v,j_v\in \{3,4,5,6,7\}$$iv,jvź{3,4,5,6,7} such that v is not incident with adjacent $$i_v$$iv-cycles and $$j_v$$jv-cycles, then the total chromatic number of graph G is $$\varDelta +1$$Δ+1.