Almost self-complementary factors of complete bipartite graphs

Abstract A complete bipartite graph without one edge, K n,m , is called almost complete bipartite graph. A graph K 2n+1,2m+1 that can be decomposed into two isomorphic factors with a given diameter d is called d - isodecomposable . We prove that K 2n+1, 2m+1 is d -isodecomposable only if d = 3, 4, 5, 6 or ∞ and completely determine all d -isodecomposable almost complete bipartite graphs for each diameter. For d = ∞ we, moreover, present all classes of possible disconnected factors.