Related papers: Efficiently list-edge coloring multigraphs asympto…

Let $G=G(n)$ be a graph on $n$ vertices with maximum degree $\Delta=\Delta(n)$. Assign to each vertex $v$ of $G$ a list $L(v)$ of colors by choosing each list independently and uniformly at random from all $k$-subsets of a color set…

We train an agent to navigate in 3D environments using a hierarchical strategy including a high-level graph based planner and a local policy. Our main contribution is a data driven learning based approach for planning under uncertainty in…

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovasz local lemma in probabilistic combinatorics. We show that the conclusion of the Lovasz local lemma holds for dependency…

The individualization-refinement paradigm provides a strong toolbox for testing isomorphism of two graphs and indeed, the currently fastest implementations of isomorphism solvers all follow this approach. While these solvers are fast in…

A probabilistic version of the Weisfeiler-Leman algorithm for computing the coherent closure of a colored graph is suggested. The algorithm is Monte Carlo and runs in time $ O(n^{1+\omega}\log^2 n) $, where $ n $ is the number of vertices…

In the multicoloring problem, also known as ($a$:$b$)-coloring or $b$-fold coloring, we are given a graph G and a set of $a$ colors, and the task is to assign a subset of $b$ colors to each vertex of G so that adjacent vertices receive…

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…

Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if it succeeds in…

Let $P_k$ be a path, $C_k$ a cycle on $k$ vertices, and $K_{k,k}$ a complete bipartite graph with $k$ vertices on each side of the bipartition. We prove that (1) for any integers $k, t>0$ and a graph $H$ there are finitely many subgraph…

Brooks' Theorem [R. L. Brooks, On Colouring the Nodes of a Network, Proc. Cambridge Philos. Soc.} 37:194-197, 1941] states that every graph $G$ with maximum degree $\Delta$, has a vertex-colouring with $\Delta$ colours, unless $G$ is a…

Graph coloring is fundamental to distributed computing. We give the first sub-logarithmic distributed algorithm for coloring cluster graphs. These graphs are obtained from the underlying communication network by contracting nodes and edges,…

Given a large social or information network, how can we partition the vertices into sets (i.e., colors) such that no two vertices linked by an edge are in the same set while minimizing the number of sets used. Despite the obvious practical…

This paper proves that a wide class of local search algorithms extend as is to the fully dynamic setting with an adaptive adversary, achieving an amortized $\tilde{O}(1)$ number of local-search steps per update. A breakthrough by Moser…

In this paper we investigate the extent to which the Lov\'asz Local Lemma (an important tool in probabilistic combinatorics) can be adapted for the measurable setting. In most applications, the Lov\'asz Local Lemma is used to produce a…

We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are: (i) the mean and variance…

We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or…

A perfect matching M in an edge-colored complete bipartite graph K_{n,n} is rainbow if no pair of edges in M have the same color. We obtain asymptotic enumeration results for the number of rainbow matchings in terms of the maximum number of…

We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but all its restricted subhypergraphs with edges of size at least 3 are 2-colorable. This disproves a bold conjecture of Keszegh and the author, and can be considered…

Call a colouring of a graph \emph{distinguishing} if the only automorphism of this graph which preserves said colouring is the identity. Let $H$ be an arbitrary graph. We say that a graph $G$ is \emph{$H$-free} if $G$ does not contain an…

The Lovasz Local Lemma due to Erdos and Lovasz is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events,…