A solution to a conjecture on the rainbow connection number

TL;DR

This paper proves a conjecture regarding the existence of connected graphs with specified rainbow connection and strong rainbow connection numbers, confirming the conditions under which such graphs can be constructed.

Contribution

It provides a proof that for given positive integers, graphs exist with prescribed rainbow and strong rainbow connection numbers satisfying the conjecture's conditions.

Findings

The conjecture is true.

Existence of graphs with specified rainbow connection numbers.

Conditions for the values of $rc(G)$ and $src(G)$ are confirmed.

Abstract

For a graph $G$, Chartrand et al. defined the rainbow connection number $rc(G)$ and the strong rainbow connection number $src(G)$ in "G. Charand, G.L. John, K.A. Mckeon, P. Zhang, Rainbow connection in graphs, Mathematica Bohemica, 133(1)(2008) 85-98". They raised the following conjecture: for two given positive $a$ and $b$, there exists a connected graph $G$ such that $rc(G)=a$ and $src(G)=b$ if and only if $a=b\in\{1,2\}$ or $ 3\leq a\leq b$". In this short note, we will show that the conjecture is true.