Scenery Reconstruction on Finite Abelian Groups

We consider the question of when a random walk on a finite abelian group with a given step distribution can be used to reconstruct a binary labeling of the elements of the group, up to a shift. Matzinger and Lember (2006) give a sufficient condition for reconstructibility on cycles. While, as we show, this condition is not in general necessary, our main result is that it is necessary when the length of the cycle is prime and larger than 5, and the step distribution has only rational probabilities. We extend this result to other abelian groups.Comment: 16 pages, 2 figure

Path-cordial abelian groups

A labeling of the vertices of a graph by elements of any abelian group A induces a labeling of the edges by summing the labels of their endpoints. Hovey defined the graph G to be A-cordial if it has such a labeling where the vertex labels and the edge labels are both evenly-distributed over A in a technical sense. His conjecture that all trees T are A-cordial for all cyclic groups A remains wide open, despite significant attention. Curiously, there has been very little study of whether Hovey’s conjecture might extend beyond the class of cyclic groups. We initiate this study by analyzing the larger class of finite abelian groups A such that all path graphs are A-cordial. We conjecture a complete characterization of such groups, and establish this conjecture for various infinite families of groups as well as for all groups of small order

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 $G[C_4]$