On a Generalized Matching Problem Arising in Estimating the Eigenvalue Variation of Two Matrices

It is shown that if G is a graph having vertices P1, P2, ..., Pn, Q1, Q2, ..., Qn and satisfying some conditions, then there is a permutation σ of {1, 2, ..., n} such that there is a path, for i = 1, 2, ..., n connecting Pi with Qσ(i) having a length at most {n/2}. This is used to prove a theorem having applications in eigenvalue variation estimation.