The minimum number of detours in graphs

Abstract

A longest path in a graph is called a detour. It is easy to see that a connected graph of minimum degree at least $2$ and order at least $4$ has at least $4$ detours. We prove that if the number of detours in such a graph of order at least $9$ is odd, then it is at least $9,$ and this lower bound can be attained for every order. Thus the possibilities $3,$ $5$ and $7$ are excluded. Two open problems are posed.