Related papers: Optimal $L(1,2)$-edge Labeling of Infinite Octagon…

A $k$-L(2,1)-labeling of a graph is a function from its vertex set into the set $\{0,...,k\}$, such that the labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices of distance 2 are different. It is known…

Let $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ be a simple graph, an $L(2,1)$-labeling of $\mathcal{G}$ is an assignment of labels from nonnegative integers to vertices of $\mathcal{G}$ such that adjacent vertices get labels which differ by…

The enhanced power graph of a finite group $G$ is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. An $L( 2,1)$-labeling of graph $\Gamma$…

For any two non-negative integers h and k, h > k, an L(h, k)-colouring of a graph G is a colouring of vertices such that adjacent vertices admit colours that at least differ by h and vertices that are two distances apart admit colours that…

A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance…

Given a finite or infinite graph $G$ and positive integers $\ell, h_1, h_2, h_3$, an $L(h_1, h_2, h_3)$-labelling of $G$ with span $\ell$ is a mapping $f: V(G) \rightarrow \{0, 1, 2, \ldots, \ell\}$ such that, for $i = 1, 2, 3$ and any $u,…

An $L(3,2,1)$-labeling of a graph $G$ is an assignment $f$ of nonnegative integers to vertices such that $\vert f(x)-f(y)\vert > 3-\mbox{dist}_G(x,y)$ for every pair $x,y$ of vertices of $G$, where $\mbox{dist}_G(x,y)$ denotes the distance…

Let $G$ be a connected graph. The edge-connectivity of $G$, denoted by $\lambda(G)$, is the minimum number of edges whose removal renders $G$ disconnected. Let $\delta(G)$ be the minimum degree of $G$. It is well-known that $\lambda(G) \leq…

We extend a result of Griggs and Yeh about the maximum possible value of the L(2,1)-labeling number of a graph in terms of its maximum degree to oriented graphs. We consider the problem both in the usual definition of the oriented…

An L(2,1)-labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ to the set of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ and $y$ are adjacent and $|f(x)-f(y)|\ge 1$ if $x$ and $y$ are at distance 2, for all…

Let $G$ be a finite simple graph on the vertex set $V(G) = \{x_1, \ldots, x_n\}$ and $I(G) \subset K[V(G)]$ its edge ideal, where $K[V(G)]$ is the polynomial ring in $x_1, \ldots, x_n$ over a field $K$ with each ${\rm deg} x_i = 1$ and…

A good edge-labelling of a simple, finite graph is a labelling of its edges with real numbers such that, for every ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. In this paper we prove that any graph on…

One of the most famous applications of Graph Theory is in the field of Channel Assignment Problems. There are varieties of graph colouring concepts that are used for different requirements of frequency assignments in communication channels.…

In this paper, we study the $L(2, 1)$-Labeling of the Mycielski and the iterated Mycielski of graphs in general. For a graph $G$ and all $t\geq 1$, we give sharp bounds for $\lambda(M^t(G))$ the $L(2, 1)$-labeling number of the $t$-th…

We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their…

Let $\lambda(G)$ be the smallest number of vertices that can be removed from a non-empty graph $G$ so that the resulting graph has a smaller maximum degree. Let $\lambda_{\rm e}(G)$ be the smallest number of edges that can be removed from…

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. The lexicographic product of…

Let $G$ be a nontrivial connected graph of order $n$ and $k$ an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such…

Edge-girth-regular graphs (abbreviated as $egr$ graphs) are a class of highly regular graphs. More specifically, for integers $v$, $k$, $g$ and $\lambda$ an $egr(v,k,g,\lambda)$ graph is a $k$-regular graph with girth $g$ on $v$ vertices…

Given a graph $G$, a \textit{$k$-total difference labeling} of the graph is a total labeling $f$ from the set of edges and vertices to the set $\{1, 2, \cdots k\}$ satisfying that for any edge $\{u,v\}$, $f(\{u,v\})=|f(u)-f(v)|$. If $G$ is…