Sufficient conditions are derived for the determination of E-optimal designs in the class D(v,b1,b2,k1,k2) of incomplete block designs for v treatments in b1 blocks of size k1 each and b2 blocks of size k2 each. Some constructions for E-optimal designs that satisfy the sufficient conditions obtained here are given. In particular, it is shown that E-optimal designs in D(v,b1,b2,k1,k2) can be ... [Show full abstract] constructed by augmenting b2 blocks, with k2 − k1 extra plots each, of a BIBD(v,b = b1 + b2,k1,λ) and GDD(v,b = b1 + b2,k1,λ1,λ2). It is also shown that equireplicate E-optimal designs in D(v,b1,b2,k1,k2) can be constructed by combining disjoint blocks of BIBD(v,b,k1,λ) and GDD(v,b,k1,λ1,λ2) into larger blocks. As applications of the construction techniques, several infinite series of E-optimal designs with small block sizes differing by at most two are given. Lower bounds for the A-efficiency are derived and it is found that A-efficiency exceeds 99% for v ⩾ 10, and at least 97.5% for 5 ⩽v < 10.