Related papers: Uniquely D-colourable digraphs with large girth

We prove that for every digraph $D$ and every choice of positive integers $k$, $\ell$ there exists a digraph $D^*$ with girth at least $\ell$ together with a surjective acyclic homomorphism $\psi\colon D^*\to D$ such that: (i) for every…

\qquad A \emph{coloring} of a digraph $D=(V,E)$ is a coloring of its vertices following the rule: Let $uv$ be an arc in $D$. If the tail $u$ is colored first, then the head $v$ should receive a color different from that of $u$. The…

We prove that for every oriented graph $D$ and every choice of positive integers $k$ and $\ell$, there exists an oriented graph $D^*$ along with a surjective homomorphism $\psi\colon V(D^*) \to V(D)$ such that: (i) girth$(D^*) \geq\ell$;…

It is known (Bollob\'{a}s (1978); Kostochka and Mazurova (1977)) that there exist graphs of maximum degree $\Delta$ and of arbitrarily large girth whose chromatic number is at least $c \Delta / \log \Delta$. We show an analogous result for…

An acyclic homomorphism of a digraph $C$ to a digraph $D$ is a function $\rho\colon V(C)\to V(D)$ such that for every arc $uv$ of $C$, either $\rho(u)=\rho(v)$, or $\rho(u)\rho(v)$ is an arc of $D$ and for every vertex $v\in V(D)$, the…

Let G be a plane graph with maximum face size D. If all faces of G with size four or more are vertex disjoint, then G has a cyclic coloring with D+1 colors, i.e., a coloring such that all vertices incident with the same face receive…

Let $G$ be a graph and c a proper k-coloring of G, i.e. any two adjacent vertices u and v have different colors c(u) and c(v). A proper k-coloring is a b-coloring if there exists a vertex in every color class that contains all the colors in…

A vertex coloring of a strong digraph $D$ is a \emph{strong vertex-monochromatic connection coloring (SVMC-coloring)} if for every pair $u, v$ of vertices in $D$ there exists an $(u,v)$-path having all its internal vertices of the same…

A graph is called uniquely distinguishing colorable if there is only one partition of vertices of the graph that forms distinguishing coloring with the smallest possible colors. In this paper, we study the unique colorability of the…

A graph is $(d_1, \ldots, d_k)$-colorable if its vertex set can be partitioned into $k$ nonempty subsets so that the subgraph induced by the $i$th part has maximum degree at most $d_i$ for each $i\in\{1, \ldots, k\}$. It is known that for…

A proper vertex-colouring of a simple graph $G$ is said to be odd if, for every non-isolated vertex $v$ of $G$, some colour appears an odd number of times in the neighbourhood of $v$. We show that if $G$ embeds in the torus, then it admits…

A graph is $(d_1, ..., d_r)$-colorable if its vertex set can be partitioned into $r$ sets $V_1, ..., V_r$ so that the maximum degree of the graph induced by $V_i$ is at most $d_i$ for each $i\in \{1, ..., r\}$. For a given pair $(g, d_1)$,…

An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$…

We consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any…

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing chromatic number $\chi_{D}(G)$ of $G$ is…

The distinguishing number $D(G)$ of a graph $G$ is the smallest number of colors that is needed to color the vertices of $G$ such that the only color preserving automorphism is the identity. For infinite graphs $D(G)$ is bounded by the…

In this note, we introduce and study a new version of neighbour-distinguishing arc-colourings of digraphs. An arc-colouring $\gamma$ of a digraph $D$ is proper if no two arcs with the same head or with the same tail are assigned the same…

Let $c$ be an edge-colouring of a graph $G$ such that for every vertex $v$ there are at least $d \ge 2$ different colours on edges incident to $v$. We prove that $G$ contains a properly coloured path of length 2d or a properly coloured…

A {\em kernel by properly colored paths} of an arc-colored digraph $D$ is a set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected by a properly colored directed path in $D$, and (ii) every vertex outside $S$ can…

An {\it odd $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing an odd number of times on its neighborhood. This concept was introduced very recently by Petru\v sevski and \v Skrekovski…