A note on the neighbour-distinguishing index of digraphs

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 colour. For each vertex $u$ of $D$, we denote by $S_\gamma^-(u)$ and $S_\gamma^+(u)$ the sets of colours that appear on the incoming arcs and on the outgoing arcs of $u$, respectively. An arc colouring $\gamma$ of $D$ is \emph{neighbour-distinguishing} if, for every two adjacent vertices $u$ and $v$ of $D$, the ordered pairs $(S_\gamma^-(u),S_\gamma^+(u))$ and $(S_\gamma^-(v),S_\gamma^+(v))$ are distinct. The neighbour-distinguishing index of $D$ is then the smallest number of colours needed for a neighbour-distinguishing arc-colouring of $D$.We prove upper bounds on the neighbour-distingui\-shing index of various classes of digraphs

Uniquely Distinguishing Colorable Graphs

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 distinguishing coloring of a graph and its applications in computing the distinguishing chromatic number of disconnected graphs. We introduce two families of uniquely distinguishing colorable graphs, namely type 1 and type 2, and show that every disconnected uniquely distinguishing colorable graph is the union of two isomorphic graphs of type 2. We obtain some results on bipartite uniquely distinguishing colorable graphs and show that any uniquely distinguishing $n$-colorable tree with $n \geq 3$ is a star graph. For a connected graph $G$, we prove that $\chi_D(G\cup G)=\chi_D(G)+1$ if and only if $G$ is uniquely distinguishing colorable of type 1. Also, a characterization of all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) = \chi_{D}(G) = k$, where $k=n-2, n-1, n$, is given in this paper. Moreover, we determine all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = \ell$, where $\ell=n-1, n, n+1$. Finally, we investigate the family of connected graphs $G$ with $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = 3$

Asymmetric colorings of products of graphs and digraphs

We extend results about asymmetric colorings of finite Cartesian products of graphs to strong and direct products of graphs and digraphs. On the way we shorten proofs for the existence of prime factorizations of finite digraphs and characterize the structure of the automorphism groups of strong and direct products. The paper ends with results on asymmetric colorings of Cartesian products of finite and infinite digraphs.http://www.elsevier.com/locate/dam2020-08-15hj2019Mathematics and Applied Mathematic

Distinguishing numbers and distinguishing indices of oriented graphs

A distinguishing r-vertex-labelling (resp. r-edge-labelling) of an undirected graph G is a mapping λ from the set of vertices (resp. the set of edges) of G to the set of labels {1,. .. , r} such that no non-trivial automorphism of G preserves all the vertex (resp. edge) labels. The distinguishing number D(G) and the distinguishing index D (G) of G are then the smallest r for which G admits a distinguishing r-vertex-labelling or r-edge-labelling, respectively. The distinguishing chromatic number D χ (G) and the distinguishing chromatic index D χ (G) are defined similarly, with the additional requirement that the corresponding labelling must be a proper colouring. These notions readily extend to oriented graphs, by considering arcs instead of edges. In this paper, we study the four corresponding parameters for oriented graphs whose underlying graph is a path, a cycle, a complete graph or a bipartite complete graph. In each case, we determine their minimum and maximum value, taken over all possible orientations of the corresponding underlying graph, except for the minimum values for unbalanced complete bipartite graphs K m,n with m = 2, 3 or 4 and n > 3, 6 or 13, respectively, or m ≥ 5 and n > 2 m − m 2 , for which we only provide upper bounds

Feedback vertex sets in (directed) graphs of bounded degeneracy or treewidth

We study the minimum size $f$ of a feedback vertex set in directed and undirected $n$-vertex graphs of given degeneracy or treewidth. In the undirected setting the bound $\frac{k-1}{k+1}n$ is known to be tight for graphs with bounded treewidth $k$ or bounded odd degeneracy $k$. We show that neither of the easy upper and lower bounds $\frac{k-1}{k+1}n$ and $\frac{k}{k+2}n$ can be exact for the case of even degeneracy. More precisely, for even degeneracy $k$ we prove that $f 0$, there exists a $k$-degenerate graph for which $f\geq \frac{3k-2}{3k+4}n -\epsilon$. For directed graphs of bounded degeneracy $k$, we prove that $f\leq\frac{k-1}{k+1}n$ and that this inequality is strict when $k$ is odd. For directed graphs of bounded treewidth $k\geq 2$, we show that $f \leq \frac{k}{k+3}n$ and for every $\epsilon>0$, there exists a $k$-degenerate graph for which $f\geq \frac{k-2\lfloor\log_2(k)\rfloor}{k+1}n -\epsilon$. Further, we provide several constructions of low degeneracy or treewidth and large $f$.Comment: 19 pages, 7 figures, 2 table