Odd Cycles and Hilbert Functions of Their Toric Rings

Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each 1 ≤ s ≤ 5 , an O-sequence ( h 0 , h 1 , … , h 2 s − 1 ) ∈ Z ≥ 0 2 s satisfying the properties that (i) h 0 ≤ h 1 ≤ ⋯ ≤ h s − 1 , (ii) h 2 s − 1 = h 0 , h 2 s − 2 = h 1 and (iii) h 2 s − 1 − i = h i + ( − 1 ) i , 2 ≤ i ≤ s − 1 , can be the h-vector of a Cohen-Macaulay standard G-domain.