On common edges in optimal solutions to traveling salesman and other optimization problems

Abstract

We prove first two extensions of Smith's theorem: (i) the number of partitions of a graph into a spanning tree and a perfect matching is even, (ii) the number of partitions of a graph into a spanning tree and a hamiltonian cycle containing a given edge is even. Then we consider three optimization problems: traveling salesman, minimum perfect matching and minimum spanning tree. Using (i) and (ii) we show that for any pair of them, the corresponding optimal solutions must have some common edges.