# Total rainbow connection of digraphs

## TL;DR

This paper investigates the total rainbow connection number in directed graphs, extending existing concepts from undirected graphs to digraphs, and provides results for various classes like tournaments and cactus digraphs.

## Contribution

It introduces the concept of total rainbow connection for digraphs and presents new results for biorientations, tournaments, and cactus digraphs.

## Key findings

- Results on total rainbow connection number for biorientations
- Findings for tournaments and cactus digraphs
- Extension of rainbow connection concepts to digraphs

## Abstract

An edge-coloured path is rainbow if its edges have distinct colours. For a connected graph $G$, the rainbow connection number (resp. strong rainbow connection number) of $G$ is the minimum number of colours required to colour the edges of $G$ so that, any two vertices of $G$ are connected by a rainbow path (resp. rainbow geodesic). These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. Krivelevich and Yuster generalised this concept to the vertex-coloured setting. Similarly, Liu, Mestre and Sousa introduced the version which involves total-colourings. Dorbec, Schiermeyer, Sidorowicz and Sopena extended the concept of the rainbow connection to digraphs. In this paper, we consider the (strong) total rainbow connection number of digraphs. Results on the (strong) total rainbow connection number of biorientations of graphs, tournaments and cactus digraphs are presented.