Related papers: Exact bounds for distributed graph colouring
The fastest algorithms for edge coloring run in time $2^m n^{O(1)}$, where $m$ and $n$ are the number of edges and vertices of the input graph, respectively. For dense graphs, this bound becomes $2^{\Theta(n^2)}$. This is a somewhat unique…
We show that any randomised Monte Carlo distributed algorithm for the Lov\'asz local lemma requires $\Omega(\log \log n)$ communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the…
We present a randomized distributed algorithm that computes a $\Delta$-coloring in any non-complete graph with maximum degree $\Delta \geq 4$ in $O(\log \Delta) + 2^{O(\sqrt{\log\log n})}$ rounds, as well as a randomized algorithm that…
In this paper we study fractional coloring from the angle of distributed computing. Fractional coloring is the linear relaxation of the classical notion of coloring, and has many applications, in particular in scheduling. It was proved by…
It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without…
The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any n-vertex, connected graph is at least (1+o(1)) n…
By prior work, it is known that any distributed graph algorithm that finds a maximal matching requires $\Omega(\log^* n)$ communication rounds, while it is possible to find a maximal fractional matching in $O(1)$ rounds in bounded-degree…
We give an almost complete characterization of the hardness of $c$-coloring $\chi$-chromatic graphs with distributed algorithms, for a wide range of models of distributed computing. In particular, we show that these problems do not admit…
We consider the communication complexity of finding an approximate maximum matching in a graph in a multi-party message-passing communication model. The maximum matching problem is one of the most fundamental graph combinatorial problems,…
A graph $G$ has maximal local edge-connectivity $k$ if the maximum number of edge-disjoint paths between every pair of distinct vertices $x$ and $y$ is at most $k$. We prove Brooks-type theorems for $k$-connected graphs with maximal local…
The proper connection number $pc(G)$ of a connected graph $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one path in $G$ such that no two…
We present a new algorithm for finding large independent sets in $3$-colorable graphs with small $1$-sided threshold rank. Specifically, given an $n$-vertex $3$-colorable graph whose uniform random walk matrix has at most $r$ eigenvalues…
The paper considers the NP-hard graph vertex coloring problem, which differs from traditional problems in which it is required to color vertices with a given (or minimal) number of colors so that adjacent vertices have different colors. In…
The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a…
A $k$-coloring of a graph is an assignment of integers between $1$ and $k$ to vertices in the graph such that the endpoints of each edge receive different numbers. We study a local variation of the coloring problem, which imposes further…
Graph coloring is fundamental to distributed computing. We give the first general treatment of the coloring of virtual graphs, where the graph $H$ to be colored is locally embedded within the communication graph $G$. Besides generalizing…
We present a new technique to efficiently sample and communicate a large number of elements from a distributed sampling space. When used in the context of a recent LOCAL algorithm for $(\operatorname{degree}+1)$-list-coloring (D1LC), this…
We study the {edge-coloring} problem in the message-passing model of distributed computing. This is one of the most fundamental and well-studied problems in this area. Currently, the best-known deterministic algorithms for (2Delta…
The problem of finding paths in temporal graphs has been recently considered due to its many applications. In this paper we consider a variant of the problem that, given a vertex-colored temporal graph, asks for a path whose vertices have…
The Colour Refinement procedure and its generalisation to higher dimensions, the Weisfeiler-Leman algorithm, are central subroutines in approaches to the graph isomorphism problem. In an iterative fashion, Colour Refinement computes a…