Abstract:

In this paper, by using the topological graphs and the edge-neighbor graphs of tiling, we give a sufficient
and necessary condition of two tiles of normal tiling having a same collection of neighborhoods. And it is proved that there exist impossibly three tiles with a same collection of neighborhoods in every normal tiling via the condition. Therefore, an open problem of Grünbaum is solved  in some sense.

CLC Number: 

• 05B45

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