相关论文: Minimum Number of Colors: the Turk's Head Knots Ca…

The Hadwiger--Nelson problem is about determining the chromatic number of the plane (CNP), defined as the minimum number of colours needed to colour the plane so that no two points of distance 1 have the same colour. In this paper we…

We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most…

The minimum sum coloring problem with bundles was introduced by Darbouy and Friggstad (SWAT 2024) as a common generalization of the minimum coloring problem and the minimum sum coloring problem. During their presentation, the following open…

Given a knot, we develop methods for finding the braid representative that minimizes the number of simple walks. Such braids lead to an efficient method for computing the colored Jones polynomial of $K$, following an approach developed by…

We call the minimum order of any complete graph so that for any coloring of the edges by $k$ colors it is impossible to avoid a monochromatic or rainbow triangle, a Mixed Ramsey number. For any graph $H$ with edges colored from the above…

The multi-fold chromatic number of the plane $\chi_m$ is the smallest number of colors $k$, sufficient to color each point of the Euclidean plane in exactly $m$ colors, so that for any pair of points at a unit distance from each other, two…

The \emph{Ramsey multiplicity constant} of a graph $H$ is the limit as $n$ tends to infinity of the minimum density of monochromatic labeled copies of $H$ in a $2$-edge colouring of $K_n$. Fox and Wigderson recently identified a large…

The slicing degree of a knot $K$ is defined as the smallest integer $k$ such that $K$ is $k$-slice in $\#^n \overline{\mathbb{CP}^2}$ for some $n$. In this paper, we establish bounds for the slicing degrees of knots using Rasmussen's…

In 1959, Goodman determined the minimum number of monochromatic triangles in a complete graph whose edge set is two-coloured. Goodman also raised the question of proving analogous results for complete graphs whose edge sets are coloured…

A path in an edge-colored graph is called {\em rainbow} if no two edges of it are colored the same. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k\leq \ell$, the {\em rainbow $k$-connection number} $rc_k(G)$ of $G$ is…

In this paper I study a variant of the general vertex coloring problem called precoloring. Specifically, I study graph precolorings, by developing new theory, for characterizing the minimal non-extensible precolorings. It is interesting per…

Let $G$ be a simple graph. The dominated coloring of $G$ is a proper coloring of $G$ such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of $G$ is called the dominated…

An $n$-crossing is a point in the projection of a knot where $n$ strands cross so that each strand bisects the crossing. An \"ubercrossing projection has a single $n$-crossing and a petal projection has a single $n$-crossing such that there…

The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree…

The chromatic sum of a graph is the smallest sum of colors among all proper colorings with natural numbers. The strength is the minimum number of colors needed to achieve the chromatic sum. We construct for each positive integer k a tree…

Let $G = (V,E)$ be a graph of order $n$ with chromatic number $\chi(G) = k$, let $S \subset V$ and let $C_0$ be a $k$-coloring of the induced subgraph $G[S]$. The coloring $C_0$ is called an extendable coloring, if $C_0$ can be extended to…

The lattice stick number $s_L(K)$ of a knot $K$ is defined to be the minimal number of straight line segments required to construct a stick presentation of $K$ in the cubic lattice. In this paper, we find an upper bound on the lattice stick…

In this paper, we study the following two hypercube coloring problems: Given $n$ and $d$, find the minimum number of colors, denoted as ${\chi}'_{d}(n)$ (resp. ${\chi}_{d}(n)$), needed to color the vertices of the $n$-cube such that any two…

An edge-colored graph $G$ is said to be rainbow connected if between each pair of vertices there exists a path which uses each color at most once. The rainbow connection number, denoted by $rc(G)$, is the minimum number of colors needed to…

In this paper, we settle the open complexity status of interval constrained coloring with a fixed number of colors. We prove that the problem is already NP-complete if the number of different colors is 3. Previously, it has only been known…