Related papers: Sperner type lemma for quadrangulations

We consider a generalization of the classic Sperner lemma. This lemma states that every Sperner coloring of a triangulation of a simplex contains a fully colored simplex. We found a weaker assumption than Sperner's coloring. It is also…

Fr\'ed\'eric Meunier's question about a multicolored Sperner lemma is addressed, leaving the question of connectivity for the color hypergraphs of such a multicolored simplex. Sperner's lemma asserts the existence of a simplex using all the…

Sperner's lemma is a statement about labeled triangulations of a simplex. McLennan and Tourky (2007) provided a novel proof of Sperner's Lemma by examining volumes of simplices in a triangulation under time-linear simplex-linear…

We discuss coloring and partitioning questions related to Sperner's Lemma, originally motivated by an application in hardness of approximation. Informally, we call a partitioning of the $(k-1)$-dimensional simplex into $k$ parts, or a…

We consider a generalization of Sperner's lemma for a triangulation $T$ of $(m+1)$-discs $D$ whose vertices are colored in $n+2$ colors. A proper coloring of $T$ on the boundary of $D$ determines a simplicial mapping $f:S^m \to S^n$ and the…

It is considered a special, convex variant of Sperner lemma type .

Attending to an open problem in the literature stated by Mirzakhani and Vondr\'ak, we give a lower bound of the number of non-monochromatic simplices for Sperner labelings of the vertices of a triangulation of a given $ k$-simplex with…

The proof of Brouwer's fixed-point theorem based on Sperner's lemma is often presented as an elementary combinatorial alternative to advanced proofs based on algebraic topology. The goal of this note is to show that: (i) the combinatorial…

We show, without using the Four Color Theorem, that for each planar triangulation, the number of its proper vertex colorings by 4 colors is a determinant and thus can be calculated in a polynomial time. In particular, we can efficiently…

We give a simple and complete description of those convex lattice polygons in the plane that can be dissected into lattice triangles of integer area. A new version of Sperner's Lemma plays a central role.

We consider Brouwer's fixed point theorem and Sperner's lemma in one dimension. We present a proof of the Brouwer theorem using the Sperner lemma, and vice versa. However, we also show that they are not equivalent, because the Sperner lemma…

Given a so called ''Sperner coloring'' of a triangulation of the $D$-dimensional simplex, Sperner's lemma guarantees the existence of a rainbow simplex, i.e. a simplex colored by all $D+1$ colors. However, finding a rainbow simplex was the…

A graph embedded in a surface with all faces of size 4 is known as a quadrangulation. We extend the definition of quadrangulation to higher dimensions, and prove that any graph G which embeds as a quadrangulation in the real projective…

We prove an extension of the well-known combinatorial-topological lemma of E. Sperner to the case of infinite-dimensional cubes. It is obtained as a corollary to an infinitary extension of the Lebesgue Covering Dimension Theorem.

We establish a "neighborhood" variant of the cubical KKM lemma and the Lebesgue covering theorem and deduce a discretized version which is a "neighborhood" variant of Sperner's lemma on the cube. The main result is the following: for any…

We answer a question posed by T. Gallai in 1969 concerning criticality in Sperner's lemma, listed as Problem 9.14 in the collection of Jensen and Toft [Graph coloring problems, John Wiley & Sons, Inc., New York, 1995]. Sperner's lemma…

We investigate Sperner's labelings of $H^\pi_{k,q}$, the hypergraph whose hyperedges are facets of the edgewise triangulation of a $(k-1)$-simplex defined by a permutation $\pi\in \mathbb{S}_{k-1}$. Mirzakhani and Vondr\' ak showed that the…

Proving for triangulations an extended version of the 4-colour theorem by induction, we manage to exclude the case which led to the failure of Kempe's attempted proof. The new idea is to claim the existence of a "nice" 4-colouring, in which…

We establish a simple generalization of a known result in the plane. The simplices in any pure simplicial complex in R^d may be colored with d+1 colors so that no two simplices that share a (d-1)-facet have the same color. In R^2 this says…

This paper provides a positive answer to the question of Mirzakhani and Vondrak that asks if there is a Sperner-admissible labeling of the simplex-lattice hypergraph such that each hyperedge uses at most 2 colors.