Note on group distance magic graphs G[C4]G[C_4]

A \emph{group distance magic labeling} or a \gr-distance magic labeling of a graph $G(V,E)$ with $|V | = n$ is an injection $f$ from $V$ to an Abelian group \gr of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of every vertex $x \in V$ is equal to the same element \mu \in \gr, called the magic constant. In this paper we will show that if $G$ is a graph of order $n=2^{p}(2k+1)$ for some natural numbers $p$, $k$ such that \deg(v)\equiv c \imod {2^{p+1}} for some constant $c$ for any $v\in V(G)$, then there exists an \gr-distance magic labeling for any Abelian group \gr for the graph $G[C_4]$. Moreover we prove that if \gr is an arbitrary Abelian group of order $4n$ such that \gr \cong \zet_2 \times\zet_2 \times \gA for some Abelian group \gA of order $n$, then exists a \gr-distance magic labeling for any graph $G[C_4]$

Decompositions of Complete Graphs Into Kayak Paddles

A canoe paddle is a cycle attached to an end-vertex of a path. It was shown by Truszczynski that all canoe paddles are graceful and therefore decompose complete graphs. A kayak paddle is a pair of cycles joined by a path. We prove that the complete graph K<sub>2n+1</sub> is decomposable into kayak paddles with <i>n</i> edges whenever at least one of its cycles is eve.DOI : http://dx.doi.org/10.22342/jims.0.0.17.39-4

Distance Magic Graphs - a Survey

Let <i>G = (V;E)</i> be a graph of order n. A bijection <i>f : V → {1, 2,...,n} </i>is called <i>a distance magic labeling </i>of G if there exists a positive integer k such that <i>Σ f(u) = k </i> for all <i>v ε V</i>, where <i>N(v)</i> is the open neighborhood of v. The constant k is called the magic constant of the labeling f. Any graph which admits <i>a distance magic labeling </i>is called a distance magic graph. In this paper we present a survey of existing results on distance magic graphs along with our recent results,open problems and conjectures.DOI : http://dx.doi.org/10.22342/jims.0.0.15.11-2

α2-labeling of graphs

We show that if a graph G on n edges allows certain special type of rosy labeling (a.k.a. rho;-labeling), called α2-labeling, then for any positive integer k the complete graph K2nk+1 can be decomposed into copies of G. This notion generalizes the α-labeling introduced in 1967 by A. Rosa

Testing Samples of Three Independent Groups

A method of testing triples consisting of samples of three independent groups such that no two samples of the same group are tested at the same time, no sample is tested with another one more than once and after each test one sample of the triple is included in the following triple is presented