INTERPOLATION THEOREM FOR THE NUMBER OF GENERALIZED END-VERTICES OF SPANNING-TREES

Abstract

Schuster has proved the following interpolation theorem. If a graph G contains spanning trees having exactly m and n end-vertices, with m < n, then for every integer p, m < p < n, G contains a spanning tree having exactly p end-vertices. To extend this theorem we generalize the end-vertex by defining the k end-vertex, where the end-vertex of G is the 1-end-vertex of G. Then we prove that the number of k-end-vertices of spanning trees of a graph G has the interpolation property for every positive integer k.