Minimum permanents on two faces of the polytope of doubly stochastic matrices

AbstractWe consider the minimum permanents and minimising matrices on the faces of the polytope of doubly stochastic matrices whose nonzero entries coincide with those of, respectively,Um,n=InJn,mJm,n0mandVm,n=InJn,mJm,nJm,m.Here Jr,s denotes the r×s matrix all of whose entries are 1, In is the identity matrix of order n and 0m is the m×m zero matrix.We conjecture that Vm,n is cohesive but not barycentric for 1<n<m+m and that it is not cohesive for n⩾m+m. We prove that it is cohesive for 1<n<m+m and not cohesive for n⩾2m and confirm the conjecture computationally for n