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In this report, we survey results on distance magic graphs and some closely related graphs. A distance magic labeling of a graph G with magic constant k is a bijection l from the vertex set to {1, 2, . . . , n}, such that for every vertex x Σ l(y) = k,y∈NG(x)
where NG(x) is the set of vertices of G adjacent to x. If the graph G has a distance magic...
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Citations
... One of the most important results for D-magic labelings is in [13,14] where it is shown that for a particular graph, the magic constant is unique and is determined by its fractional domination number. For more results, refer to survey articles [15,16]. ...
In this paper, we define D-magic labelings for oriented graphs where D is a distance set. In particular, we label the vertices of the graph with distinct integers {1,2,…,|V(G)|} in such a way that the sum of all the vertex labels that are a distance in D away from a given vertex is the same across all vertices. We give some results related to the magic constant, construct a few infinite families of D-magic graphs, and examine trees, cycles, and multipartite graphs. This definition grew out of the definition of D-magic (undirected) graphs. This paper explores some of the symmetries we see between the undirected and directed version of D-magic labelings.
... The concept of fair scheduling was independently studied by Miller et al. [31] in 2003 under the name 1-vertex magic and by Sugeng et al. [42] under the name distance magic labeling. For recent surveys on distance magic labeling, see [1,36]. The Fair-NET problem for a general multiset S was first studied by O'Neal and Slater [32]. ...
A \emph{fair competition}, based on the concept of envy-freeness, is a non-eliminating competition where each contestant (team or individual player) may not play against all other contestants, but the total difficulty for each contestant is the same: the sum of the initial rankings of the opponents for each contestant is the same. Similar to other non-eliminating competitions like the Round-robin competition or the Swiss-system competition, the winner of the fair competition is the contestant who wins the most games. The {\sc Fair Non-Eliminating Tournament} ({\sc Fair-NET}) problem can be used to schedule fair competitions whose infrastructure is known. In the {\sc Fair-NET} problem, we are given an infrastructure of a tournament represented by a graph $G$ and the initial rankings of the contestants represented by a multiset of integers $S$. The objective is to decide whether $G$ is \emph{$S$-fair}, i.e., there exists an assignment of the contestants to the vertices of $G$ such that the sum of the rankings of the neighbors of each contestant in $G$ is the same constant $k\in\mathbb{N}$. We initiate a study of the classical and parameterized complexity of {\sc Fair-NET} with respect to several central structural parameters motivated by real world scenarios, thereby presenting a comprehensive picture of it.
... One of the most important results for D-magic labeling is in [4,12] where it is shown that for a particular graph, the magic constant is unique and is determined by its fractional domination number. For more results, refer to survey articles in [3] and [13]. ...
For a set of distances D, a graph G on n vertices is said to be D-magic if there exists a bijection and a constant k such that for any vertex x, where is the D-neighbourhood set of x. In this paper we utilize spectra of graphs to characterize strongly regular graphs which are D-magic, for all possible distance sets D. In addition, we provide necessary conditions for distance regular graphs of diameter 3 to be {1}-magic.
... One of the most important results for D-magic labeling is in [12,4] where it is shown that for a particular graph, the magic constant is unique and is determined by its fractional domination number. For more results, refer to survey articles in [3] and [13]. ...
A graph $G$ on $n$ vertices is said to be distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , n\}$ and a constant $k$ such that for any vertex $x$, $\sum_{y\in N(x)} f(y) = k$, where $N(x)$ is the set of all neighbours of $x$. In this paper we utilize spectra of graphs to characterize strongly regular graphs admitting distance magic labelings. In addition, we proved that a distance regular graph of diameter 3 is distance magic only if it is primitive.
... Until now, the most important result for distance magic labeling is in [11,2] where it is shown that for a particular graph, the magic constant is unique and is determined by its fractional domination number. For more results, please refer to survey articles in [1] and [14]. ...
A graph $G$ is said to be distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , v\}$ and a constant {\sf k} such that for any vertex $x$, $\sum_{y\in N(x)} f(y) ={\sf k}$, where $N_(x)$ is the set of all neighbours of $x$. In this paper we shall study distance magic labelings of graphs obtained from four graph products: cartesian, strong, lexicographic, and cronecker. We shall utilise magic rectangle sets and magic column rectangles to construct the labelings.
... The concept of distance magic labeling was originally introduced by Vilfred [8]. For an overview of known results on distance magic labeling we refer to Arumugam et al. [1] and Rupnow [7]. Two distance magic graphs of order 7 with magic constants 9 and 7 respectively are shown in Figure 1. ...
Let \(G=(V,E)\) be a graph on n vertices. A bijection \(f: V \rightarrow \{1,2,\ldots , n\}\) is called a nearly distance magic labeling of G if there exists a positive integer k such that \(\sum _{x \in N(v)} f(x)=k \ or \ k+1\) for every \(v \in V\). The constant k is called a magic constant of the graph and any graph which admits such a labeling is called a nearly distance magic graph. In this paper we present several basic results on nearly distance magic graphs and compute the magic constant k in terms of the fractional total domination number of the graph.
A bijective mapping $f: V(G) \rightarrow \left\{1,2,\ldots,n\right\}$ is called a \emph{Distance Magic Labeling (DML) of $G$} if ~ ${\sum_{v \in N(u)}} f(v) $ is a constant for all $u\in V(G)$ where $G$ is a simple graph of order $n$ and $N(u)$ = $\{v\in V(G):$ $uv\in E(G)\}$. Graph $G$ is called a \emph{Distance Magic Graph (DMG)} if it has a DML, otherwise it is called a \emph{Non-Distance Magic (NDM) graph}. In 1996, Vilfred proposed a conjecture that cylindrical grid graphs $P_m \Box C_n$ are NDM for $m \geq 2$, $n \geq 3$ and $m,n\in\mathbb{N}$. Recently, the authors could prove the conjecture for the case when $m$ is even by introducing neighbourhood chains of Type-1 (NC-T1) and Type-2 (NC-T2). In this paper, they introduce neighbourhood chains of Type-3 (NC-T3) and using them completely settle the conjecture and also identify families of NDM graphs.
Let \(G=(V,E)\) be a graph and \(\varGamma \) be an abelian group both of order n. For \(D \subset \{0,1,\ldots , diam(G)\}\), the D
-distance neighbourhood of a vertex v in G is defined to be the set \(N_D(v)=\{x \in V \ | \ d(x,v) \in D\}\). A bijection \(f: V \rightarrow \varGamma \) is called a \((\varGamma , D)\)
-distance magic labeling of G if there exists an \(\alpha \in \varGamma \) such that \(\sum _{x \in N_D(v)} f(x)=\alpha \) for every \(v \in V\). In this paper we study \((\varGamma , D)\)-distance magic labeling of the graph \(C_n^r\) for \(D=\{d\}\). We obtain \((\varGamma , \{d\})\)-distance magic labelings of \(C_n^r\) with respect to certain classes of abelian groups. We also obtain necessary conditions for existence of such labelings.
