Related papers: On the k-edge magic graphs

Let $k \in \mathbb{N}$ and $c \in \mathbb{Z}_k$, where $\mathbb{Z}_1=\mathbb{Z}$. A graph $G=(V(G),E(G))$ is said to be $c$-sum $k$-magic if there is a labeling $\ell:E(G) \rightarrow \mathbb{Z}_k \setminus \{0\}$ such that $\sum_{u \in…

Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these…

A $(2,1)$-total labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ and the edge set $E(G)$ to the set $\{0,1,...,k\}$ of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ is a vertex and $y$ is an edge incident…

The bend-number b(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of…

An antimagic labelling of a graph $G = (V,E)$ is a bijection from $E$ to $\{1,2, \ldots, |E|\}$, such that all vertex-sums are pairwise distinct, where the vertex-sum of each vertex is the sum of labels over edges incident to this vertex. A…

A graph drawn in the plane is called k-quasi-planar if it does not contain k pairwise crossing edges. It has been conjectured for a long time that for every fixed k, the maximum number of edges of a k-quasi-planar graph with n vertices is…

A $k$-antimagic labeling of a graph $G$ is an injection from $E(G)$ to $\{1,2,\dots,|E(G)|+k\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $u$ is the sum of the labels assigned to edges incident to $u$.…

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said…

Let $G=(V,E)$ be an $n$-vertex graph with $m$ edges. A function $f : V \cup E \rightarrow \{1, \ldots, n+m\}$ is an edge-magic labeling of $G$ if $f$ is bijective and, for some integer $k$, we have $f(u)+f(v)+f(uv) = k$ for every edge $uv…

A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and any list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that…

We study the energy per vertex in regular graphs. For every k, we give an upper bound for the energy per vertex of a k-regular graph, and show that a graph attains the upper bound if and only if it is the disjoint union of incidence graphs…

A mapping $l : E(G) \rightarrow A$, where $A$ is an abelian group which written additively, is called a labeling of the graph $G$. For every positive integer $h \geqslant 2$, a graph $G$ is said to be zero-sum $h$-magic if there is an edge…

A graph $G$ is called edge-magic if there is a bijective function $f$ from the set of vertices and edges to the set $\{1,2,\ldots,|V(G)|+|E(G)|\}$ such that the sum $f(x)+f(xy)+f(y)$ for any $xy$ in $E(G)$ is constant. Such a function is…

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,…

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said…

A subset $R\subseteq V(G)$ of a graph $G$ is a general position set if any triple set $R_0$ of $R$ is non-geodesic in $G$, that is, no vertex of $R_0$ lies on any geodesic between the other two vertices of $R_0$ in $G$. Let $\mathcal{R}$ be…

Given a graph $G = (V,E)$, a set $S \subset V$ is called a $k$-\emph{metric generator} for $G$ if any pair of different vertices of $G$ is distinguished by at least $k$ elements of $S$. A graph is $k$-\emph{metric dimensional} if $k$ is the…

A $\Gamma$-distance magic labeling of a graph $G=(V,E)$ with $|V | = n$ is a bijection $\ell$ from $V$ to an Abelian group $\Gamma$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}\ell(y)$ of every vertex $x \in V$ is equal to the…

Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a…

A finite, simple and undirected graph $G = (V, E)$ with $p$ vertices and $q$ edges is said to be a $k$-geometric mean graph for a positive integer $k$ if there is an injection $\psi :V(G)\to \{k,k+1,\dots,k+q\}$ such that, when each edge…