Related papers: Simple Sublinear Algorithms for $(\Delta+1)$ Verte…

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $\Delta$ can be {\em edge colored} using at most $\Delta + 1$ different colors [Diskret.~Analiz, '64]. Vizing's original proof is algorithmic and shows that such…

The property of spatial mixing and strong spatial mixing in spin systems has been of interest because of its implications on uniqueness of Gibbs measures on infinite graphs and efficient approximation of counting problems that are otherwise…

We consider the following extension of the concept of adjacent strong edge colourings of graphs without isolated edges. Two distinct vertices which are at distant at most $r$ in a graph are called $r$-adjacent. The least number of colours…

All Colors Shortest Path problem defined on an undirected graph aims at finding a shortest, possibly non-simple, path where every color occurs at least once, assuming that each vertex in the graph is associated with a color known in…

A celebrated result of Johansson in graph theory states that every triangle-free graph of maximum degree $\Delta$ can be properly colored with $O(\Delta/\ln\Delta)$ colors, improving upon the "greedy bound" of $\Delta+1$ coloring in general…

Given a proper edge coloring $\varphi$ of a graph $G$, we define the palette $S_{G}(v,\varphi)$ of a vertex $v \in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check s(G)$ of $G$ is the minimum…

Given an undirected graph $G = (V,E)$ with a set $V$ of vertices and a set $E$ of edges, the minimum sum coloring problem (MSCP) is to find a legal vertex coloring of $G$, using colors represented by natural numbers $1, 2, . . .$ such that…

We present a randomized algorithm which takes as input an undirected graph $G$ on $n$ vertices with maximum degree $\Delta$, and a number of colors $k \geq (8/3 + o_{\Delta}(1))\Delta$, and returns -- in expected time…

A colouring of a graph $G=(V,E)$ is a mapping $c\colon V\to \{1,2,\ldots\}$ such that $c(u)\neq c(v)$ for every two adjacent vertices $u$ and $v$ of $G$. The {\sc List $k$-Colouring} problem is to decide whether a graph $G=(V,E)$ with a…

This paper presents massively parallel computation (MPC) algorithms in the strongly sublinear memory regime (aka, scalable MPC) for orienting and coloring graphs as a function of its subgraph density. Our algorithms run in $poly(\log\log…

We consider the problem of sampling and approximately counting an arbitrary given motif $H$ in a graph $G$, where access to $G$ is given via queries: degree, neighbor, and pair, as well as uniform edge sample queries. Previous algorithms…

We consider a variation of the spectral sparsification problem where we are required to keep a subgraph of the original graph. Formally, given a union of two weighted graphs $G$ and $W$ and an integer $k$, we are asked to find a $k$-edge…

Given an edge-colored graph, the goal of the proportional fair matching problem is to find a maximum weight matching while ensuring proportional representation (with respect to the number of edges) of each color. The colors may correspond…

The paper presents techniques to derive upper bounds for the mean time to recover from a single fault for self-stabilizing algorithms in the message passing model. For a new Delta+1-coloring algorithm we analytically derive a bound for the…

We give a randomized algorithm that properly colors the vertices of a triangle-free graph G on n vertices using O(\Delta(G)/ log \Delta(G)) colors, where \Delta(G) is the maximum degree of G. The algorithm takes O(n\Delta2(G)log\Delta(G))…

We study Markov chains for randomly sampling $k$-colorings of a graph with maximum degree $\Delta$. Our main result is a polynomial upper bound on the mixing time of the single-site update chain known as the Glauber dynamics for planar…

A spectral sparsifier of a graph $G$ is a sparser graph $H$ that approximately preserves the quadratic form of $G$, i.e. for all vectors $x$, $x^T L_G x \approx x^T L_H x$, where $L_G$ and $L_H$ denote the respective graph Laplacians.…

We introduce a generalization of the well known graph (vertex) coloring problem, which we call the problem of \emph{component coloring of graphs}. Given a graph, the problem is to color the vertices using minimum number of colors so that…

We show that, for every $k \ge 2$, every $k$-uniform hypergaph of degree $\Delta$ and girth at least $5$ is efficiently $(1+o(1) )(k-1) (\Delta / \ln \Delta )^{ 1/(k-1) } $-list colorable. As an application (and to the best of our…

Irregular computations on unstructured data are an important class of problems for parallel programming. Graph coloring is often an important preprocessing step, e.g. as a way to perform dependency analysis for safe parallel execution. The…