Tessellation and g-tessellation of circulants, Q6, and Qt6 Original Research Article

Let C subset of or equal to Rn be a pointed cone generated by rational vectors. A subset H of integral vectors in C is said to be a Hilbert basis for C if all integer vectors in C can be expressed as nonnegative integer combinations of vectors in H. A tessellation of C is a partition of C into unimodular subcones whose generators are from H. The existence of such a tessellation has consequences similar to Carathéodoryʹs theorem in linear algebra. It is an open question whether such a tessellation exists for all C. In this paper we show that tessellation (or its generalization) exists for cones generated by circulant matrices, Q6, and Qt6.