Related papers: Colourful Simplicial Depth

We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this…

For a degree sequence $d:d_1\geq \cdots \geq d_n$, we consider the smallest chromatic number $\chi_{\min}(d)$ and the largest chromatic number $\chi_{\max}(d)$ among all graphs with degree sequence $d$. We show that if $d_n\geq 1$, then…

Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$. Let $L_{j}$ denote the $j$-Laplacian acting on real $j$-cochains of $X$ and let $\mu_{j}(X)$ denote its minimal eigenvalue. We study the…

We construct $d$-dimensional pure simplicial complexes and pseudo-manifolds (without boundary) with $n$ vertices whose combinatorial diameter grows as $c_d n^{d-1}$ for a constant $c_d$ depending only on $d$, which is the maximum possible…

The Cyclic Coloring Conjecture asserts that the vertices of every plane graph with maximum face size D can be colored using at most 3D/2 colors in such a way that no face is incident with two vertices of the same color. The Cyclic Coloring…

The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge-coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by K\H{o}nig, $d$ colors suffice if the graph is bipartite. We investigate…

Archdeacon (1987) proved that graphs embeddable on a fixed surface can be $3$-coloured so that each colour class induces a subgraph of bounded maximum degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no…

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…

Let $K_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a $2$-colouring of the edges of $K_{\mathbb{N}}$ in which every monochromatic path has density~$0$.…

For a proper immersed minimal disk in $\bf{R}^N$ with quadratic area growth, we show that any harmonic function whose negative part grows at a slow sub-linear rate is constant. This leads to a higher codimensional Bernstein theorem for…

A simplicial complex of dimension $d-1$ is said to be balanced if its graph is $d$-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a…

The diameter of a strongly connected $d$-dimensional simplicial complex is the diameter of its dual graph. We provide a probabilistic proof of the existence of $d$-dimensional simplicial complexes with diameter $ (\frac{1}{d \cdot d!} -…

Let $P$ be a set of $n$ points in $d$-dimensions. The simplicial depth, $\sigma_P(q)$ of a point $q$ is the number of $d$-simplices with vertices in $P$ that contain $q$ in their convex hulls. The simplicial depth is a notion of data depth…

Recently Lau-Jeyaseeli-Shiu-Arumugam introduced the concept of the "Sudoku colourings" of graphs -- partial $\chi(G)$-colourings of $G$ that have a unique extension to a proper $\chi(G)$-colouring of all the vertices. They introduced the…

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…

Let L_1, ..., L_d be pairwise disjoint collections of lines in a d-dimensional vector space over some field. If the collections are sufficiently generic we prove that there exists a d-colouring of the set of multijoints J such that for each…

We present an explicit family of hypergraphs with arbitrarily large uniformity and chromatic number that admit realizations in both geometric and number-theoretic settings. As an application, we give a new proof of a theorem of Chen, Pach,…

The investigation of colour symmetries for periodic and aperiodic systems consists of two steps. The first concerns the computation of the possible numbers of colours and is mainly combinatorial in nature. The second is algebraic and…

Motivated both by recently introduced forms of list colouring and by earlier work on independent transversals subject to a local sparsity condition, we use the semi-random method to prove the following result. For any function $\mu$…

Motivated by algorithmic applications, Kun, O'Brien, Pilipczuk, and Sullivan introduced the parameter linear chromatic number as a relaxation of treedepth and proved that the two parameters are polynomially related. They conjectured that…