Related papers: $2$-colourability of the maximum ranked elements o…

In this paper we will show the existence of a face $2$-colourable biembedding of the complete graph onto an orientable surface where each face is a cycle of a fixed length $k$, for infinitely many values of $k$. In particular, under certain…

Stan Wagon asked the following in 2000. Is every zonohedron face 3-colorable when viewed as a planar map? An equivalent question, under a different guise, is the following: is the arrangement graph of great circles on the sphere always…

We give an explicit combinatorial description of the two-dimensional faces of both the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ of a partially ordered set $P$. Using these descriptions, we show that for any…

We prove a simple formula for arbitrary cluster variables in the marked surfaces model. As part of the formula, we associate a labeled poset to each tagged arc, such that the associated $F$-polynomial is a weighted sum of order ideals. Each…

To the integral symplectic group Sp(2g,Z) we associate two posets of which we prove that they have the Cohen-Macaulay property. As an application we show that the locus of marked decomposable principally polarized abelian varieties in the…

We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of…

Let $S$ be a finite set of geometric objects partitioned into classes or \emph{colors}. A subset $S'\subseteq S$ is said to be \emph{balanced} if $S'$ contains the same amount of elements of $S$ from each of the colors. We study several…

We show that double cosets of the infinite symmetric group with respect to some special subgroups admit natural structures of semigroups. We interpret elements of such semigroups in combinatorial terms (chips, colored graphs,…

We exhibit several posets arising from commutative algebra, order theory, tropical convexity as potential face posets of tropical polyhedra, and we clarify their inclusion relations. We focus on monomial tropical polyhedra, and deduce how…

A plane graph is l-facially k-colourable if its vertices can be coloured with k colours such that any two distinct vertices on a facial segment of length at most l are coloured differently. We prove that every plane graph is 3-facially…

We consider orientation-preserving actions of finite groups $G$ on pairs $(S^3, \Sigma)$, where $\Sigma$ denotes a compact connected surface embedded in $S^3$. In a previous paper, we considered the case of closed, necessarily orientable…

Let $G=(V,E)$ be a simple graph and $(2k+1)$ be a prime integer. Let each vertex of $G$ be colored using one of the $(2k+1)$ colors, say $R_1,R_2,...,R_{2k+1}$. If every vertex has an equal number of neighbors of each color, then the…

A modular or distributive lattice is `diamond-colored' if its order diagram edges are colored in such a way that, within any diamond of edges, parallel edges have the same color. Such lattices arise naturally in combinatorial representation…

In 1986 Stanley associated to a poset the order polytope. The close interplay between its combinatorial and geometric properties makes the order polytope an object of tremendous interest. Double posets were introduced in 2011 by Malvenuto…

In this paper we introduce the concepts of the distinguishing number and the distinguishing chromatic number of a poset. For a distributive lattice $L$ and its set $Q_L$ of join-irreducibles, we use classic lattice theory to show that any…

We consider the problem of $2$-coloring geometric hypergraphs. Specifically, we show that there is a constant $m$ such that any finite set of points in the plane $\mathcal{S} \subset {\mathbb R}^2$ can be $2$-colored such that every…

This is the first in a sequence of three papers in which we prove the following generalization of Thomassen's 5-choosability theorem: Let $G$ be a finite graph embedded on a surface of genus $g$. Then $G$ can be $L$-colored, where $L$ is a…

For any Coxeter system $(W,S)$ of rank $n$, we introduce an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples $(I,w,J)$, where $I$ and…

We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set,…

Two vertex colorings of a graph are Kempe equivalent if they can be transformed into each other by a sequence of switchings of two colors of vertices. It is PSPACE-complete to determine whether two given vertex $k$-colorings of a graph are…