Related papers: Breaking small automorphisms by list colourings
A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many…
For a graph $G$, a $k$-coloring $c:V(G)\to \{1,2,\ldots, k\}$ is called distinguishing, if the only automorphism $f$ of $G$ with the property $c(v)=c(f(v))$ for every vertex $v\in G$ (color-preserving automorphism), is the identity. In this…
A well-studied concept is that of the total chromatic number. A proper total colouring of a graph is a colouring of both vertices and edges so that every pair of adjacent vertices receive different colours, every pair of adjacent edges…
We introduce the {\it endomorphism distinguishing number} $D_e(G)$ of a graph $G$ as the least cardinal $d$ such that $G$ has a vertex coloring with $d$ colors that is only preserved by the trivial endomorphism. This generalizes the notion…
We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph K_{s,t} which admits only the identity automorphism. In particular this allows us to determine the distinguishing number of the…
We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have…
A graph $G$ is said to be $d$-distinguishable if there is a labeling of the vertices with $d$ labels so that only the trivial automorphism preserves the labels. The smallest such $d$ is the distinguishing number, Dist($G$). A subset of…
A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms of $G$ can preserve it. The distinguishing number of $G$, denoted by $D(G)$, is the minimum number of colors required for such a coloring, and the…
The $k$-th exact-distance graph, of a graph $G$ has $V(G)$ as its vertex set, and $xy$ as an edge if and only if the distance between $x$ and $y$ is (exactly) $k$ in $G$. We consider two possible extensions of this notion for signed graphs.…
Let G be a simple, finite, connected, and undirected graph. The middle graph M(G) of G is obtained from the subdivision graph S(G) after joining pairs of subdivided vertices that lie on adjacent edges of G and the central graph C(G) of G is…
A proper vertex-colouring of a graph G is said to be locally identifying if for any pair u,v of adjacent vertices with distinct closed neighbourhoods, the sets of colours in the closed neighbourhoods of u and v are different. We show that…
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…
An edge-coloring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Lehner, Pil\'{s}niak, and Stawiski proved that all connected regular graphs except $K_2$ admit an asymmetric edge-coloring with…
The distinguishing number of a graph $G$ is the smallest $k$ such that $G$ admits a $k$-colouring for which the only colour-preserving automorphism of $G$ is the identity. We determine the distinguishing number of finite $4$-valent…
We give elementary proofs of the following two theorems on automorphisms of a finite group G: (1) An automorphism of G is inner if and only if it extends to an automorphism of every finite group containing G. (2) There exists a finite…
Call a colouring of a graph \emph{distinguishing} if the only automorphism of this graph which preserves said colouring is the identity. Let $H$ be an arbitrary graph. We say that a graph $G$ is \emph{$H$-free} if $G$ does not contain an…
An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part…
A colouring of a graph $G=(V,E)$ is a mapping $c\colon V\to \{1,2,\ldots\}$ such that $c(u)\neq c(v)$ for every two adjacent vertices $u$ and $v$ of $G$. The {\sc List $k$-Colouring} problem is to decide whether a graph $G=(V,E)$ with a…
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval…
A circulant (di)graph is a (di)graph on n vertices that admits a cyclic automorphism of order n. This paper provides a survey of the work that has been done on finding the automorphism groups of circulant (di)graphs, including the…