turkmath.org

In 1968, Golomb and Welch conjectured that Z n cannot be tiled by Lee spheres with a fixed radius r ≥ 2 for dimension n ≥ 3. Besides its own interest in discrete geometry and coding theory, if we restrict this conjecture to the lattice tiling case it is also equivalent to the nonexistence of abelian Cayley graphs archiving the Moore-like bound for the cardinality of vertices. A question on the existence of abelian Cayley graphs achieving this upper bound except for some trivial examples has been proposed independently in the context of graph theory for many years. In this talk, I will first give a brief survey of known results. Then I will sketch a proof on the nonexistence of lattice tilings of Z n by Lee spheres of radius 2 with n ≥ 3. As a consequence, we will see that the order of any abelian Cayley graph of diameter 2 and degree larger than 5 cannot meet the abelian Cayley Moore-like bound. This talk is based on a recent joint work with Ka Hin Leung.