Related papers: Identifying codes in line digraphs

For nonnegative integers $k$ and $l$, let $\mathscr{D}(k,l)$ denote the family of digraphs in which every vertex has either indegree at most $k$ or outdegree at most $l$. In this paper we prove that the edges of every digraph in…

Let $G$ be a simple finite connected graph. The line graph $L(G)$ of graph $G$ is the graph whose vertices are the edges of $G$, where $ef \in E(L(G))$ when $e \cap f \neq \emptyset$. Iteratively, the higher order line graphs are defined…

Given $\alpha>0$ and an integer $\ell\geq5$, we prove that every sufficiently large $3$-uniform hypergraph $H$ on $n$ vertices in which every two vertices are contained in at least $\alpha n$ edges contains a copy of $C_\ell^{-}$, a tight…

A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors…

A set S of vertices of a graph G is a dominating set of G if every vertex u of G is either in S or it has a neighbour in S. In other words, S is dominating if the sets S\cap N[u] where u \in V(G) and N[u] denotes the closed neighbourhood of…

The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…

For $0\leq \ell <k$, a Hamiltonian $\ell$-cycle in a $k$-uniform hypergraph $H$ is a cyclic ordering of the vertices of $H$ in which the edges are segments of length $k$ and every two consecutive edges overlap in exactly $\ell$ vertices. We…

Let $r,k,\ell$ be integers such that $0\le\ell\le\binom{k}{r}$. Given a large $r$-uniform hypergraph $G$, we consider the fraction of $k$-vertex subsets which span exactly $\ell$ edges. If $\ell$ is 0 or $\binom{k}{r}$, this fraction can be…

A graph $G$ is said to be a $(k,\ell)$-graph if its vertex set can be partitioned into $k$ independent sets and $\ell$ cliques. It is well established that the recognition problem for $(k,\ell)$-graphs is NP-complete whenever $k \geq 3$ or…

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 $k$-uniform hypergraph with $n$ vertices is an $(n,k,\ell)$-omitting system if it does not contain two edges whose intersection has size exactly $\ell$. If in addition it does not contain two edges whose intersection has size greater than…

Identifying and locating-dominating codes have been studied widely in circulant graphs of type $C_n(1,2,3,\dots, r)$ over the recent years. In 2013, Ghebleh and Niepel studied locating-dominating and identifying codes in the circulant…

The digraph chromatic number of a directed graph $D$, denoted $\chi_A(D)$, is the minimum positive integer $k$ such that there exists a partition of the vertices of $D$ into $k$ disjoint sets, each of which induces an acyclic subgraph. For…

Let $n>k>\ell$ be positive integers. We say a $k$-uniform hypergraph $\mathcal{H}$ contains a Hamilton $(\ell,k-\ell)$-cycle if there is a partition $(L_0,R_0,L_1,R_1,\ldots,L_{t-1},R_{t-1})$ of $V(\mathcal{H})$ with $|L_i|=\ell$,…

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$…

The dichromatic number $\dic(D)$ of a digraph $D$ is the least integer $k$ such that $D$ can be partitioned into $k$ directed acyclic digraphs. A digraph is $k$-dicritical if $\dic(D) = k$ and each proper subgraph $D'$ of $D$ satisfies…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. We examine the effects on…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. In this paper, we investigate…

A graph is $k$-planar if it can be drawn in the plane so that each edge is crossed at most $k$ times. Typically, the class of 1-planar graphs is among the most investigated graph families within the so-called "beyond planar graphs". A…

The girth of a graph $G$ is the length of a shortest cycle of $G$. Jiang (JCT-B, 2001) showed that every graph $G$ with girth at least $2\ell+1$ and minimum degree at least $k/\ell$ contains every tree $T$ with $k$ edges whose maximum…