The minimum number of detours in a connected graph of minimum degree three

Abstract

A longest path in a graph is called a detour. Denote by $a(k,n)$ the minimum number of detours in a connected graph with minimum degree $k$ and order $n,$ and denote by $b(k,n)$ the minimum odd number of detours in such a graph. X. Zhan has posed the problem of determining $a(k,n)$ and $b(k,n).$ It is known that $a(2,n)=4$ for $n\ge 4$ and $b(2,n)=9$ for $n\ge 9.$ In this paper we prove that $a(3,n)=36$ for $n\ge 18,$ $a(k,n)\le (k!)^2$ for $n\ge k^2+2k+3$ and $b(3,n)\le 225$ for $n\ge 11.$ We also pose several related unsolved problems.