Double-Wheel Graphs Are Graceful

We present the first polynomial time construction procedure for generating graceful double-wheel graphs. A graph is graceful if its vertices can be labeled with distinct integer values from {0, ..., e}, where e is the number of edges, such that each edge has a unique value corresponding to the absolute difference of its endpoints. Graceful graphs have a range of practical application domains, including in radio astronomy, X-ray crystallography, cryptography, and experimental design. Various families of graphs have been proven to be graceful, while others have only been conjectured to be. In particular, it has been conjectured that so-called double-wheel graphs are graceful. A double-wheel graph consists of two cycles of N nodes connected to a common hub. We prove this conjecture by providing the first construction for graceful double-wheel graphs, for any N > 3, using a framework that combines streamlined constraint reasoning with insights from human computation. We also use this framework to provide a polynomial time construction for diagonally ordered magic squares.