相关论文: Sufficient conditions for a digraph to admit a $(1…
Given an integer $\ell\ge 1$, a $(1,\le \ell)$-identifying code in a digraph is a dominating subset $C$ of vertices such that all distinct subsets of vertices of cardinality at most $\ell$ have distinct closed in-neighbourhood within $C$.…
An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\M(G)$ denotes the minimum size of an identifying code of a graph $G$, it was…
We call a subset $C$ of vertices of a graph $G$ a $(1,\leq \ell)$-identifying code if for all subsets $X$ of vertices with size at most $\ell$, the sets $\{c\in C |\exists u \in X, d(u,c)\leq 1\}$ are distinct. The concept of identifying…
An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the…
An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We study the edge-identifying code problem, i.e. the identifying code…
The distinguishing index of a simple graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. It was conjectured by Pil\'sniak (2015) that for any 2-connected…
An identifying code of a closed-twin-free graph $G$ is a dominating set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhoods and $S$. It was conjectured that there exists…
Let $\delta$ and $\Delta$ be the minimum and the maximum degree of the vertices of a simple connected graph $G$, respectively. The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of…
An identifying code of a graph G is a dominating set C such that every vertex x of G is distinguished from all other vertices by the set of vertices in C that are at distance at most 1 from x. The problem of finding an identifying code of…
An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement…
An $\textit{identifying code}$ of a closed-twin-free graph $G$ is a set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhood and $S$. It was conjectured that there exists a…
We prove that, for every $\ell\geq 4$, there exists an $\ell$-vertex graph and a first order sentence having a quantifier depth at most $\ell-1$ defining the property of having an induced subgraph isomorphic to the given one. We prove that…
It is shown that any graph with maximum degree $\Delta$ in which the average degree of the induced subgraph on the set of all neighbors of any vertex exceeds $\frac{6k^2}{6k^2 + 1}\Delta + k + 6$ is either $(\Delta - k)$-colorable or…
We show that for each \ell\geq 4 every sufficiently large oriented graph G with \delta^+(G), \delta^-(G) \geq \lfloor |G|/3 \rfloor +1 contains an \ell-cycle. This is best possible for all those \ell\geq 4 which are not divisible by 3.…
An identifying open code of a graph $G$ is a set $S$ of vertices that is both a separating open code (that is, $N_G(u) \cap S \ne N_G(v) \cap S$ for all distinct vertices $u$ and $v$ in $G$) and a total dominating set (that is, $N(v) \cap S…
For a directed graph $G$, a $t$-identifying code is a subset $S\subseteq V(G)$ with the property that for each vertex $v\in V(G)$ the set of vertices of $S$ reachable from $v$ by a directed path of length at most $t$ is both non-empty and…
Mader conjectured in 1979 that an average degree of at least $3k-1$ in a graph is sufficient for the existence of a $(k+1)$-connected subgraph. The following minimum degree analogue holds: Every graph with minimum degree at least $3k-1$…
An L(2, 1)-labeling of a graph is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive numbers differed by at least 2, and vertices at distance 2 are assigned distinct numbers. The L(2, 1)-labeling…
In an undirected graph $G$, a subset $C\subseteq V(G)$ such that $C$ is a dominating set of $G$, and each vertex in $V(G)$ is dominated by a distinct subset of vertices from $C$, is called an identifying code of $G$. The concept of…
The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…