Related papers: Minimal reconstructions of a coloring

Let $X$ be a connected, locally finite graph with symmetric growth. We prove that there is a vertex coloring $\phi\colon X\to\{0,1\}$ and some $R\in\mathbb{N}$ such that every automorphism $f$ preserving $\phi$ is $R$-close to the identity…

A relational structure is homomorphism-homogeneous (HH-homogeneous for short) if every homomorphism between finite induced substructures of the structure can be extended to a homomorphism over the whole domain of the structure. Similarly, a…

A graph $H$ is {\em $p$-edge colorable} if there is a coloring $\psi: E(H) \rightarrow \{1,2,\dots,p\}$, such that for distinct $uv, vw \in E(H)$, we have $\psi(uv) \neq \psi(vw)$. The {\sc Maximum Edge-Colorable Subgraph} problem takes as…

An $(m, n)$-colored-mixed graph $G=(V, A_1, A_2,\cdots, A_m, E_1, E_2,\cdots, E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an…

We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a…

A relational structure $\mathrm{R}$ is {\em rainbow Ramsey} if for every finite induced substructure $\mathrm{C}$ of $\mathrm{R}$ and every colouring of the copies of $\mathrm{C}$ with countably many colours, such that each colour is used…

A variational model for reconstruction of damaged color images is studied, in particular in the case where only finitely many colors are admissible for the reconstructed image. An existence result and regularity properties of minimizers are…

For positive integers $N$ and $r \geq 2$, an $r$-monotone coloring of $\binom{\{1,\dots,N\}}{r}$ is a 2-coloring by $-1$ and $+1$ that is monotone on the lexicographically ordered sequence of $r$-tuples of every $(r+1)$-tuple…

For a graph $H$, the $H$-recolouring problem $\operatorname{Recol}(H)$ asks, for two given homomorphisms from a given graph $G$ to $H$, if one can get between them by a sequence of homomorphisms of $G$ to $H$ in which consecutive…

A hamiltonian coloring $c$ of a graph $G$ of order $n$ is a mapping $c$ : $V(G) \rightarrow \{0,1,2,...\}$ such that $D(u, v)$ + $|c(u) - c(v)|$ $\geq$ $n-1$, for every two distinct vertices $u$ and $v$ of $G$, where $D(u, v)$ denotes the…

An ordered graph $\mathcal{G}$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the smallest integer $N$ such that every 2-coloring of the edges of the complete ordered graph…

We show that for every finite colouring of the natural numbers there exists $a,b >1$ such that the triple $\{a,b,a^b\}$ is monochromatic. We go on to show the partition regularity of a much richer class of patterns involving exponentiation.…

Recent results show that the structural similarity of graphs can be characterized by counting homomorphisms to them: the Tree Theorem states that the well-known color-refinement algorithm does not distinguish two graphs G and H if and only…

A distinguishing coloring of a graph is a vertex coloring such that only the identity automorphism of the graph preserves the coloring. A 2-distinguishable graph is a graph which can be distinguished using 2 colors. The cost $\rho(G)$ of a…

This paper begins by exploring some old and new results about Ramsey numbers and minimum numbers of monochromatic triangles in $2$-colorings of complete graphs, both in the disjoint and non-disjoint cases. We then extend the theory, by…

The distinguishing chromatic number, $\chi_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $\varphi$, of $G$, such that the only automorphism of $G$ that preserves all colors of $\varphi$ is the identity map.…

The Hom complex ${\rm Hom}(T,G)$ of graphs is a CW-complex associated to a pair of graphs $T$ and $G$, considered in the graph coloring problem. It is known that certain homotopy invariants of ${\rm Hom}(T,G)$ give lower bounds for the…

Circular coloring is a constraints satisfaction problem where colors are assigned to nodes in a graph in such a way that every pair of connected nodes has two consecutive colors (the first color being consecutive to the last). We study…

Given a graph G, a colouring is an assignment of colours to the vertices of G so that no two adjacent vertices are coloured the same. If all colour classes have size at most t, then we call the colouring t-bounded, and the t-bounded…

For an undirected, simple, finite, connected graph $G$, we denote by $V(G)$ and $E(G)$ the sets of its vertices and edges, respectively. A function $\varphi:E(G)\rightarrow\{1,2,\ldots,t\}$ is called a proper edge $t$-coloring of a graph…