Related papers: Coloring discrete pseudomanifolds

Fix $d\ge2$ and consider a uniformly random set $P$ of $n$ points in $[0,1]^{d}$. Let $G$ be the Hasse diagram of $P$ (with respect to the coordinatewise partial order), or alternatively let $G$ be the Delaunay graph of $P$ with respect to…

We prove the existence of a finitely dependent proper colouring of the integer lattice Z^d that is fully isometry-invariant in law, for all dimensions d. Previously this was known only for d=1, while only translation-invariant examples were…

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is…

A $(d-1)$-dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs).…

An edge-colouring of a graph $G$ is said to be colour-balanced if there are equally many edges of each available colour. We are interested in finding a colour-balanced perfect matching within a colour-balanced clique $K_{2nk}$ with a…

In this short note we show that every connected $2$-edge coloured cubic graph admits an $10$-colouring. This lowers the best known upper bound for the chromatic number of connected $2$-edge coloured cubic graphs.

We consider the maximum chromatic number of hypergraphs consisting of cliques that have pairwise small intersections. Designs of the appropriate parameters produce optimal constructions, but these are known to exist only when the number of…

In [Ho] A.J. Hoffman proved a lower bound on the chromatic number of a graph in the terms of the largest and the smallest eigenvalues of its adjacency matrix. In this paper, we prove a higher dimensional version of this result and give a…

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A…

A major research area in discrete geometry is to consider the best way to partition the $d$-dimensional Euclidean space $\mathbb{R}^d$ under various quality criteria. In this paper we introduce a new type of space partitioning that is…

The defective chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $d$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has maximum degree at most $d$.…

We study the discrete graph-metric analogue of Gromov's filling area problem for the cycle graph \(C_n\). An abstract triangulation \(K\) is an isometric filling of \(C_n\) if \(\partial K=C_n\) and the graph distance between any two…

The chromatic number of a subset of Euclidean space is the minimal number of colors sufficient for coloring all points of this subset in such a way that any two points at the distance 1 have different colors. We give new upper bounds for…

An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per…

The $\!{}\bmod k$ chromatic index of a graph $G$ is the minimum number of colors needed to color the edges of $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1\!\!\pmod k$. Recently, the…

Let M be a compact Riemannian manifold with boundary. Let b>0 be the number of connected components of its boundary. For manifolds of dimension at least 3, we prove that it is possible to obtain an arbitrarily large (b+1)-th Steklov…

In a proper edge-coloring the edges of every color form a matching. A matching is induced if the end-vertices of its edges induce a matching. A strong edge-coloring is an edge-coloring in which the edges of every color form an induced…

A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors…

Fix $k \geq 3$, and let $G$ be a $k$-uniform hypergraph with maximum degree $\Delta$. Suppose that for each $l = 2, ..., k-1$, every set of l vertices of G is in at most $\Delta^{(k-l)/(k-1)}/f$ edges. Then the chromatic number of $G$ is…

We examine $t$-colourings of oriented graphs in which, for a fixed integer $k \geq 1$, vertices joined by a directed path of length at most $k$ must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the…