2-halvable Complete 4-partite Graphs

A complete 4-partite graph Km1,m2,m3,m4 is called d-halvable if it can be decomposed into two isomorphic factors of diameter d. In the class of graphs Km1,m2,m3,m4 with at most one odd part all d-halvable graphs are known. In the class of biregular graphs Km1,m2,m3,m4 with four odd parts (i.e., the graphs Km,m,m,n and Km,m,n,n) all d-halvable graphs are known as well, except for the graphs Km,m,n,n when d = 2 and n 6= m. We prove that such graphs are 2-halvable iff n,m ≥ 3. We also determine a new class of non-halvable graphs Km1,m2,m3,m4 with three or four different odd parts.