The maximum number of cycles in the complement of a tree

Abstract

In this paper we investigate trees on a fixed set of vertices whose complements contain the maximum possible number of cycles. Let T be a tree and c( T′) be the number of cycles in the complement of T. We prove that for every tree T of n⩾6 vertices and with diameter d between 4 and n−2 inclusive, there is a tree T 1 of n vertices with diameter at least d+1 so that c( T′ 1)> c( T′). We further deduce that among all trees of n⩾9 vertices, a path on n vertices has the maximum number of cycles in its complement. This settles in the affirmative a conjecture of K.B. Reid.