Related papers: Chromatic number of ordered graphs with forbidden …

An edge-ordered graph is a graph with a total ordering of its edges. A path $P=v_1v_2\ldots v_k$ in an edge-ordered graph is called increasing if $(v_iv_{i+1}) > (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$; it is called decreasing if…

A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic…

Let $\partial_H(u)$ be the set of edges incident with a vertex $u$ in the graph $H$. We say that a graph $G$ is $H$-colorable if there exist total functions $f : E(G) \rightarrow E(H)$ and $g : V(G) \rightarrow V(H)$ such that $f$ is a…

An $(m,n)$-colored mixed graph $G$ is a graph with its arcs having one of the $m$ different colors and edges having one of the $n$ different colors. A homomorphism $f$ of an $(m,n)$-colored mixed graph $G$ to an $(m,n)$-colored mixed graph…

Let $G$ be a graph embedded in a fixed surface $\Sigma$ of genus $g$ and let $L=(L(v):v\in V(G))$ be a collection of lists such that either each list has size at least five, or each list has size at least four and $G$ is triangle-free, or…

We prove that, for every function $f:\mathbb{N} \rightarrow \mathbb{N}$, there is a graph $G$ with uncountable chromatic number such that, for every $k \in \mathbb{N}$ with $k \geq 3$, every subgraph of $G$ with fewer than $f(k)$ vertices…

Let $\phi(k)$ be the minimum number of vertices in a non-$k$-choosable $k$-chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that $\phi(k) \ge 2k+2$. This bound is tight if $k$ is even. If $k$ is odd, then it is…

A graph is $k$-critical if it is $k$-chromatic but each of its proper induced subgraphs is ($k-1$)-colorable. It is known that the number of $4$-critical $P_5$-free graphs is finite, but there is an infinite number of $k$-critical…

For a graph $G$ and a tree-decomposition $(T, \mathcal{B})$ of $G$, the chromatic number of $(T, \mathcal{B})$ is the maximum of $\chi(G[B])$, taken over all bags $B \in \mathcal{B}$. The tree-chromatic number of $G$ is the minimum…

We study the existence of periodic colorings and orientations in locally finite graphs. A coloring or orientation of a graph $G$ is periodic if the resulting colored or oriented graph is quasi-transitive, meaning that $V(G)$ has finitely…

An edge-colored graph $G$ is \emph{conflict-free connected} if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$,…

If G is a graph and H is a set of subgraphs of G, then an edge-coloring of G is called H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, denoted poly_H(G), is the largest…

An ordered graph is a graph enhanced with a linear order on the vertex set. An ordered graph is a core if it does not have an order-preserving homomorphism to a proper subgraph. We say that $H$ is the core of $G$ if (i) $H$ is a core, (ii)…

We extend a recent construction concerning polychromatic colorings of hereditary hypergraph families. For every integer $h\ge 4$ we construct a $(2h-1)$-uniform hypergraph which has no polychromatic $3$-coloring, but all of whose $h$-heavy…

A graph coloring has bounded clustering if each monochromatic component has bounded size. This paper studies such a coloring, where the number of colors depends on an excluded complete bipartite subgraph. This is a much weaker assumption…

Two independent edges in ordered graphs can be nested, crossing or separated. These relations define six types of subgraphs, depending on which relations are forbidden. We refine a remark by Erd\H{o}s and Rado that every 2-coloring of the…

Let $G$ be a graph. An acyclic $k$-coloring of $G$ is a map $c:V(G)\rightarrow \{1,\dots,k\}$ such that $c(u)\neq c(v)$ for any $uv\in E(G)$ and the subgraph induced by the vertices of any two colors $i,j\in \{1,\dots,k\}$ is a forest. If…

An ordered graph $G_<$ is a graph with a total ordering $<$ on its vertex set. A monotone path of length $k$ is a sequence of vertices $v_1<v_2<\ldots<v_k$ such that $v_iv_{j}$ is an edge of $G_<$ if and only if $|j-i|=1$. A bi-clique of…

For a fixed graph $H$ on $k$ vertices, and a graph $G$ on at least $k$ vertices, we write $G\rightarrow H$ if in any vertex-coloring of $G$ with $k$ colors, there is an induced subgraph isomorphic to $H$ whose vertices have distinct colors.…

Given a graph $G$ and a target graph $H$, an $H$-coloring of $G$ is an adjacency-preserving vertex map from $G$ to $H$. By appropriate choice of $H$, these colorings can express, for instance, the independent sets or proper vertex colorings…