相关论文: Four-coloring Eulerian triangulations of the torus

Given a set $S$ of $n$ points in the plane, a \emph{radial ordering} of $S$ with respect to a point $p$ (not in $S$) is a clockwise circular ordering of the elements in $S$ by angle around $p$. If $S$ is two-colored, a \emph{colored radial…

We prove that graphs excluding a fixed immersion have bounded nonrepetitive chromatic number. More generally, we prove that if $H$ is a fixed planar graph that has a planar embedding with all the vertices with degree at least 4 on a single…

We show that there is no 4-dimensional analogue of the 5-prismatoids used in Santos' recent counterexample to the Hirsch conjecture.

DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced by Dvo\u{r}\'{a}k and Postle (2017). Recently, Huang et al. [https://doi.org/10.1016/j.amc.2019.124562] showed that planar graphs with…

In this note, we prove that any 2-coloring of the plane contains 4 points of the same color forming a rhombus with unit sides and non-unit diagonals, answering a question of Axenovich, Liu, and the second author.

Four-Color Theorem has secret in its logical proof and actual operating. In this paper we will give a proof of Four-Color Theorem based on Kuratowski's Theorem using some induction argument and give a description of the most complicated…

We consider representations of tensors as sums of decomposable tensors or, equivalently, decomposition of multilinear forms into one--forms. In this short note we show that there exists a particular finite strongly orthogonal decomposition…

A finite Euclidean set is diameter-Ramsey if, for every number of colors, some finite same-diameter witness has the property that every coloring of the witness contains a monochromatic congruent copy of the set. Frankl, Pach, Reiher and…

For the four-color theorem that has been developed over one and half centuries, all people believe it right but without complete proof convincing all1-3. Former proofs are to find the basic four-colorable patterns on a planar graph to…

A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly…

In 1965 Edmonds showed that every eulerian graph has a bi-eulerian embedding, i.e., an embedding with exactly two faces, each bounded by an euler circuit. We refine this result by giving conditions for a graph to have a bi-eulerian…

For a plane near-triangulation $G$ with the outer face bounded by a cycle $C$, let $n^\star_G$ denote the function that to each $4$-coloring $\psi$ of $C$ assigns the number of ways $\psi$ extends to a $4$-coloring of $G$. The block-count…

Triangulated surfaces are compact Riemann surfaces equipped with a conformal triangulation by equilateral triangles. In 2004, Brooks and Makover asked how triangulated surfaces are distributed in the moduli space of Riemann surfaces as the…

We show that for every g > 1 there is a compact arithmetic Riemann surface of genus g with at least 4(g-1) automorphisms, and that this lower bound is attained by infinitely many genera, the smallest being 24.

A torus-covering $T^2$-link of degree $n$ is a surface-link consisting of tori, in the form of an unbranched covering of degree $n$ over the standard torus. We focus on a torus-covering $T^2$-link of degree 3, which is determined by a pair…

We give explicit realizations with small integer coordinates for all triangulated tori with up to 12 vertices. In particular, we provide coordinate-minimal realizations in general position for all triangulations of the torus with 7, 8, 9,…

For every finite dimensional Lie group one can consider the group of all smooth loops on it, called its loop group. Such loop groups have long been studied for, among other reasons, their relations to conformal field theories and…

In this paper, we show that every $(2P_2,K_4)$-free graph is 4-colorable. The bound is attained by the five-wheel and the complement of the seven-cycle. This answers an open question by Wagon \cite{Wa80} in the 1980s. Our result can also be…

We show that every 4-dimensional torus with a linear symplectic form can be fully filled by one symplectic ball. If such a torus is not symplectomorphic to a product of 2-dimensional tori with equal sized factors, then it can also be fully…

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms…