Related papers: Coloring discrete pseudomanifolds

Discrete d-manifolds are classes of finite simple graphs which can triangulate classical manifolds but which are defined entirely within graph theory. We show that the chromatic number X(G) of a discrete d-manifold G is sandwiched between…

We give superexponential lower and upper bounds on the number of coloured $d$-dimensional triangulations whose underlying space is an oriented manifold, when the number of simplices goes to infinity and $d\geq 3$ is fixed. In the special…

Higher chromatic numbers $\chi_s$ of simplicial complexes naturally generalize the chromatic number $\chi_1$ of a graph. In any fixed dimension $d$, the $s$-chromatic number $\chi_s$ of $d$-complexes can become arbitrarily large for…

In this paper, we explore algebraic approaches to $d$-improper and $t$-clustered colourings, where the colouring constraints are relaxed to allow some monochromatic edges. Bilu [J. Comb. Theory Ser. B, 96(4):608-613, 2006] proved a…

If each edge (u,v) of a graph G=(V,E) is decorated with a permutation pi_{u,v} of k objects, we say that it has a permuted k-coloring if there is a coloring sigma from V to {1,...,k} such that sigma(v) is different from pi_{u,v}(sigma(u))…

In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d<2(k-1)log(k-1). From previous lower bounds due to Molloy and Reed,…

A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1, 2, ..., x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and…

The multi-fold chromatic number of the plane $\chi_m$ is the smallest number of colors $k$, sufficient to color each point of the Euclidean plane in exactly $m$ colors, so that for any pair of points at a unit distance from each other, two…

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a k-colorable graph with k colors so that a maximum fraction of edges are properly colored (i.e. their endpoints receive…

The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set (colour) but contained in a minimal number of colourful…

A complete $k$-coloring of a graph $G$ is a (not necessarily proper) $k$-coloring of the vertices of $G$, such that each pair of different colors appears in an edge. A complete $k$-coloring is also called connected, if each color class…

The dichromatic number and the diachromatic number are generalizations of the chromatic number and the achromatic number for digraphs considering acyclic colorings. In this paper, we determine the diachromatic number of digraphs arising…

Given an $n$-vertex graph $G$ and two positive integers $d,k \in \mathbb{N}$, the ($d,kn$)-differential coloring problem asks for a coloring of the vertices of $G$ (if one exists) with distinct numbers from 1 to $kn$ (treated as…

The colorful simplicial depth of a collection of d+1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial…

The acyclic chromatic index (or acyclic edge-chromatic number) of a graph is the least number of colors needed to properly color its edges so that none of its cycles has only two colors. We show that for a graph of max degree $\Delta$, the…

Motivated from the surrounding property of a point set in $\mathbb{R}^d$ introduced by Holmsen, Pach and Tverberg, we consider the transversal number and chromatic number of a simplicial sphere. As an attempt to give a lower bound for the…

This article is about chromatic numbers of hyperbolic surfaces. For a metric space, the $d$-chromatic number is the minimum number of colors needed to color the points of the space so that any two points at distance $d$ are of a different…

We develop a general method for constructing random manifolds and submanifolds in arbitrary dimensions. The method is based on associating colors to the vertices of a triangulated manifold, as in recent work for curves in 3-dimensional…

In this paper an improved bound on the chromatic number of the Pancake graph $P_n, n\geqslant 2$, is presented. The bound is obtained using a subadditivity property of the chromatic number of the Pancake graph. We also investigate an…