Related papers: Minimal reconstructions of a coloring

We study a reconstruction problem for colorings. Given a finite or countable set $X$, a coloring on $X$ is a function $\varphi: [X]^{2}\to \{0,1\}$, where $[X]^{2}$ is the collection of all 2-elements subsets of $X$. A set $H\subseteq X$ is…

In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper $r$-coloring $\varphi$ of a graph $G$. We investigate the problem of finding a proper $r$-coloring of $G$, which is "the…

It is consistent for every (1 <= n< omega) that (2^omega = omega_n) and there is a function (F:[omega_n]^{< omega}-> omega) such that every finite set can be written at most (2^n-1) ways as the union of two distinct monocolored sets. If GCH…

A graph \( G \) is said to be (vertex) non-repetitively colored if no simple path in \( G \) has a sequence of vertex colors that forms a repetition. Formally, a coloring \( c: V(G) \to \{1, 2, \dots, k\} \) is non-repetitive if, for every…

Let $G = (V,E)$ be a finite simple graph. Recall that a proper coloring of $G$ is a mapping $\varphi: V\to\{1,\ldots,k\}$ such that every color class induces an independent set. Such a $\varphi$ is called a semi-matching coloring if the…

We say that a vertex colouring $\varphi$ of a graph $G$ is nonrepetitive if there is no positive integer $n$ and a path on $2n$ vertices $v_{1}\ldots v_{2n}$ in $G$ such that the associated sequence of colours…

Let $f: V(G)\cup E(G)\rightarrow \{1,2,\dots,k\}$ be a non-proper total $k$-coloring of $G$. Define a weight function on total coloring as $$\phi(x)=f(x)+\sum\limits_{e\ni x}f(e)+\sum\limits_{y\in N(x)}f(y),$$ where $N(x)=\{y\in V(G)|xy\in…

A sequence is called non-repetitive if no of its subsequences forms a repetition (a sequence $r_1,r_2,\dots,r_{2n}$ such that $r_i=r_{n+i}$ for all $1\leq i \leq n$). Let $G$ be a graph whose vertices are coloured. A colouring $\varphi$ of…

For a fixed graph $H$, the reconfiguration problem for $H$-colourings (i.e. homomorphisms to $H$) asks: given a graph $G$ and two $H$-colourings $\varphi$ and $\psi$ of $G$, does there exist a sequence $f_0,\dots,f_m$ of $H$-colourings such…

We define a $P$-compelling coloring as a proper coloring of the vertices of a graph such that every subset consisting of one vertex of each color has property $P$. The $P$-compelling chromatic number is the minimum number of colors in such…

We prove that every partial function with finite domain and range can be effectively simulated through sequential colorings of graphs. Namely, we show that given a finite set $S=\{0,1,\ldots,m-1\}$ and a number $n \geq \max\{m,3\}$, any…

For a proper vertex coloring $c$ of a graph $G$, let $\varphi_c(G)$ denote the maximum, over all induced subgraphs $H$ of $G$, the difference between the chromatic number $\chi(H)$ and the number of colors used by $c$ to color $H$. We…

A complete $k$-coloring of a graph $G=(V,E)$ is an assignment $\varphi:V\to\{1,\ldots,k\}$ of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one…

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The \emph{nonrepetitive chromatic number} of a graph $G$ is the…

A {\em balanced coloring} of a graph $G$ means a triple $\{P_1,P_2,X\}$ of mutually disjoint subsets of the vertex-set $V(G)$ such that $V(G)=P_1 \uplus P_2 \uplus X$ and $|P_1|=|P_2|$. A {\em balanced decomposition} associated with the…

A ring is a graph $R$ whose vertex set can be partitioned into $k \geq 4$ nonempty sets, $X_1, \dots, X_k$, such that for all $i \in \{1,\dots,k\}$, the set $X_i$ can be ordered as $X_i = \{u_i^1, \dots, u_i^{|X_i|}\}$ so that $X_i…

We discuss representations and colorings of orthogonality hypergraphs in terms of their two-valued states interpretable as classical truth assignments. Such hypergraphs, if they allow for a faithful orthogonal representation, have quantum…

Many graph coloring proofs proceed by showing that a minimal counterexample to the theorem being proved cannot contain certain configurations, and then showing that each graph under consideration contains at least one such configuration;…

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…

We count the number of countable homogeneous colored linear orderings in $k$ colors. Relatedly, we count the number of countable $C_{n,m}$-homogeneous linear orderings. $C_{n,m}$-homogeneity is a strong homogeneity notion that approximates…