A Complete Characterization of all Magic Constants Arising from Distance Magic Graphs

A positive integer $k$ is called a magic constant if there is a graph $G$ along with a bijective function $f$ from $V(G)$ to first $|V(G)|$ natural numbers such that the weight of the vertex $w(v) = \sum_{uv \in E}f(v) =k$ for all $v \in V$. It is known that all odd positive integers greater equal $3$ and the integer powers of $2$, $2^{t}$, $t \ge 6$ are magic constants. In this paper we characterise all positive integers which are magic constants

Tetravalent distance magic graphs of small order and an infinite family of examples

A graph of order $n$ is distance magic if it admits a bijective labeling of its vertices with integers from $1$ to $n$ such that each vertex has the same sum of the labels of its neighbors. This paper contributes to the long term project of characterizing all tetravalent distance magic graphs. With the help of a computer we find that out of almost nine million connected tetravalent graphs up to order $16$ only nine are distance magic. In fact, besides the six well known wreath graphs there are only three other examples, one of each of the orders $12$, $14$ and $16$. We introduce a generalization of wreath graphs, the so-called quasi wreath graphs, and classify all distance magic graphs among them. This way we obtain infinitely many new tetravalent distance magic graphs. Moreover, the two non-wreath graphs of orders $12$ and $14$ are quasi wreath graphs while the one of order $16$ can be obtained from a quasi wreath graph of order $14$ using a simple construction due to Kov\'a\v{r}, Fron\v{c}ek and Kov\'a\v{r}ov\'a.Comment: 11 pages, 4 figure

Some distance magic graphs

A graph G = ( V , E ) , where | V | = n and | E | = m is said to be a distance magic graph if there exists a bijection from the vertex set V to the set { 1 , 2 , … , n } such that, ∑ v ∈ N ( u ) f ( v ) = k , for all u ∈ V , which is a constant and independent of u , where N ( u ) is the open neighborhood of the vertex u . The constant k is called the distance magic constant of the graph G and such a labeling f is called distance magic labeling of G . In this paper, we present new results on distance magic labeling of C n r and neighborhood expansion D n ( G ) of a graph G . Keywords: Distance magic labeling, Sigma labeling, Circulant graph