The odd girth of generalized Johnson graphs

For any non-negative integers v>k>i, the generalized Johnson graph, X=J(v,k,i), is the graph whose vertices are the k-subsets of a v-set, and where any two vertices A and B are adjacent whenever |A∩B|=i. In this note, we prove that if v≥2k and (v,k,i)≠(2k,k,0), then the odd girth of X is given by:og(X)=2⌈k−iv−2k+2i⌉+1.