Unextendible product bases and 1-factorization of complete graphs

Let f(k1,…,km)f(k1,…,km) be the minimal value of size of all possible unextendible product bases in the tensor product space ⊗i=1mCki. We have trivial lower bounds n(k1,…,km)=∑i=1m(ki-1)+1 and upper bound k1⋯kmk1⋯km. Alon and Lovász determined all cases such that f(k1,…,km)=n(k1,…,km)f(k1,…,km)=n(k1,…,km). In this paper we determine all cases such that f(k1,…,km)=k1⋯kmf(k1,…,km)=k1⋯km by presenting a sharper upper bound. We also determine several cases such that f(k1,…,km)=n(k1,…,km)+1f(k1,…,km)=n(k1,…,km)+1 by using a result on 1-factorization of complete graphs.