Related papers: On Triangles in Colored Pseudoline Arrangements
Thomassen conjectured that every triangle-free planar graph on n vertices has exponentially many 3-colorings, and proved that it has at least 2^[n^(1/12)/20000] distinct 3-colorings. We show that it has at least 2^sqrt(n/362) distinct…
The aim of this paper is to generalize the notion of the coloring complex of a graph to hypergraphs. We present three different interpretations of those complexes -- a purely combinatorial one and two geometric ones. It is shown, that most…
We propose an open question that seeks to generalise the Four Colour Theorem from two to three dimensions. As an appetiser, we show that 12 instead of four colours are both sufficient and necessary to colour every 2-complex that embeds in a…
A subspace arrangement defined by intersections of hyperplanes of the braid arrangement can be encoded by an edge colored hypergraph. It turns out that the characteristic polynomial of this type of subspace arrangement is given by a…
We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.
We generalize the notion of orthogonal latin squares to colorings of simple graphs. Two $n$-colorings of a graph are said to be \emph{orthogonal} if whenever two vertices share a color in one coloring they have distinct colors in the other…
We study 3-coloring properties of triangle-free planar graphs $G$ with two precolored 4-cycles $C_1$ and $C_2$ that are far apart. We prove that either every precoloring of $C_1\cup C_2$ extends to a 3-coloring of $G$, or $G$ contains one…
Let $H$ be a triple system with maximum degree $d>1$ and let $r>10^7\sqrt{d}\log^{2}d$. Then $H$ has a proper vertex coloring with $r$ colors such that any two color classes differ in size by at most one. The bound on $r$ is sharp in order…
Maximal planar graph refers to the planar graph with the most edges, which means no more edges can be added so that the resulting graph is still planar. The Four-Color Conjecture says that every planar graph without loops is 4-colorable.…
We consider the problem of linearly ordered (LO) coloring of hypergraphs. A hypergraph has an LO coloring if there is a vertex coloring, using a set of ordered colors, so that (i) no edge is monochromatic, and (ii) each edge has a unique…
Gy\'arf\'as and Lehel and independently Faudree and Schelp proved that in any 2-coloring of the edges of $K_{n,n}$ there exists a monochromatic path on at least $2\lceil n/2\rceil$ vertices, and this is tight. We prove a stability version…
We investigate the relationship between two kinds of vertex colorings of graphs: unique-maximum colorings and conflict-free colorings. In a unique-maximum coloring, the colors are ordered, and in every path of the graph the maximum color…
The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs with geometric representations and competitive analysis of on-line algorithms, which became apparent after the recent construction…
A well known problem from an excellent book of Lov\'asz states that any hypergraph with the property that no pair of hyperedges intersect in exactly one vertex can be properly 2-colored. Motivated by this as well as recent works of Keszegh…
It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles on a given Euclidean space has countable chromatic number, while the hypergraph of isosceles triangles in the plane does not.
We introduce the concepts of marked multi-colorings, marked chromatic polynomials, and marked (multivariate) independence series for hypergraphs. We show that the coefficients of the q-th power of the marked independence series of a…
Colored triangulations offer a generalization of combinatorial maps to higher dimensions. Just like maps are gluings of polygons, colored triangulations are built as gluings of special, higher-dimensional building blocks, such as octahedra,…
A path (cycle) in a $2$-edge-colored multigraph is alternating if no two consecutive edges have the same color. The problem of determining the existence of alternating Hamiltonian paths and cycles in $2$-edge-colored multigraphs is an…
An edge coloring of the $n$-vertex complete graph, $K_n$, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for $n$ large and every $k$…
The dichromatic and diachromatic numbers of a digraph are the minimum and maximum numbers of colors, respectively, in acyclic and complete colorings of the digraph. In this paper, we construct, for all $r \leq t$, non-symmetric digraphs…