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

TL;DR

This paper investigates the minimum number of detours (longest paths) in connected graphs with minimum degree three, providing exact and upper bound results for various graph sizes.

Contribution

It determines exact values and bounds for the minimum number of detours in graphs with minimum degree three, addressing a problem posed by X. Zhan.

Findings

Proves that a(3,n)=36 for n≥18.

Establishes that a(k,n)≤(k!)^2 for n≥k^2+2k+3.

Shows that b(3,n)≤225 for n≥11.

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.