On the construction of xor-magic graphs

A simple connected graph of order $2^n$ is defined as a xor-magic graph of power $n$ if its vertices can be labeled with vectors from $F_2^n$ in a one-to-one manner such that the sum of labels in each closed neighborhood set of vertices equals zero. In this paper, we introduce a method called the self-switching operation, which, when properly applied to an odd xor-magic graph of power $n$, generates a xor-magic graph of power $n+1$. We demonstrate the existence of a proper self-switching operation for any given odd xor-magic graph and provide a characterization of the cut space of a connected graph in the process. We also observe that the Dyck graph can be obtained from the complete graph of order 4 by applying three successive self-switching operations. Additionally, we investigate various graph products, including Cartesian, tensor, strong, lexicographical, and modular products. We observe that these products allow us to generate xor-magic graphs by selecting appropriate factor graphs. Notably, we discover that a modular product of graphs is always a xor-magic graph when the orders of its factors are powers of 2 (except for 2 itself). In the process, we realize that the Clebsch graph is the modular product of the cycle graph and the empty graph, each of order 4. By combining the self-switching operation with the modular product, we establish the existence of $k$-regular xor-magic graphs of power $n$ for all $n\ge 2$ and for all $k\in \{3,5,7,\dots ,2^n\text{-}5\}\cup \left\{2^n\text{-}1\right\}$. We also prove that there is no ($2^n\text{-}3$)-regular xor-magic graph of power $n$. Lastly, we introduce two more methods to produce xor-magic graphs. One method utilizes Cayley graphs and the other utilizes linear algebra.