Proper rainbow connection number of graphs

TL;DR

This paper studies the proper rainbow connection number of graphs, providing an improved upper bound, showing the potential for large differences from the rainbow connection number, and identifying conditions where they are equal.

Contribution

It establishes a new upper bound for the proper rainbow connection number and explores its relationship with the rainbow connection number and edge chromatic index.

Findings

Proper rainbow connection number is at most equal to the number of vertices in the graph.

The difference between proper rainbow connection number and rainbow connection number can be arbitrarily large.

Certain graph classes satisfy the equality prc(G) = χ'(G).

Abstract

A path in an edge-coloured graph is called \emph{rainbow path} if its edges receive pairwise distinct colours. An edge-coloured graph is said to be \emph{rainbow connected} if any two distinct vertices of the graph are connected by a rainbow path. The minimum $k$ for which there exists such an edge-colouring is the rainbow connection number $rc(G)$ of $G.$ Recently, Bau et al. \cite{BJJKM2018} introduced this concept with the additional requirement that the edge-colouring must be proper. %An proper edge-coloured graph is said to be \emph{properly rainbow connected} if any two distinct vertices of the graph are connected by a rainbow path. The \emph{proper rainbow connection number} of $G$, denoted by $prc(G)$, is the minimum number of colours needed in order to make it properly rainbow connected. In this paper we first prove an improved upper bound $prc(G) \leq n$ for every connected graph $G$ of order $n \geq 3.$ Next we show that the difference $prc(G) - rc(G)$ can be arbitrarily large. Finally, we present several sufficient conditions for graph classes satisfying $prc(G) = \chi'(G).$