Rainbow vertex connection of digraphs

TL;DR

This paper investigates the (strong) rainbow vertex-connection number in directed graphs, extending existing concepts from undirected graphs and analyzing specific classes like cycles, circulants, and tournaments.

Contribution

It introduces the study of rainbow vertex-connection in digraphs and provides results for various classes of digraphs, expanding the understanding of this graph parameter.

Findings

Results for biorientations of graphs

Analysis of cycle digraphs

Insights into circulant digraphs and tournaments

Abstract

An edge-coloured path is \emph{rainbow} if its edges have distinct colours. An edge-coloured connected graph is said to be \emph{rainbow connected} if any two vertices are connected by a rainbow path, and \emph{strongly rainbow connected} if any two vertices are connected by a rainbow geodesic. The (\emph{strong}) \emph{rainbow connection number} of a connected graph is the minimum number of colours needed to make the graph (strongly) rainbow connected. These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. As an extension, Krivelevich and Yuster proposed the concept of \emph{rainbow vertex-connection}. The topic of rainbow connection in graphs drew much attention and various similar parameters were introduced, mostly dealing with undirected graphs. Dorbec, Schiermeyer, Sidorowicz and Sopena extended the concept of the rainbow connection to digraphs. In this paper, we consider the (strong) rainbow vertex-connection number of digraphs. Results on the (strong) rainbow vertex-connection number of biorientations of graphs, cycle digraphs, circulant digraphs and tournaments are presented.