# Vertex-monochromatic connectivity of strong digraphs

## TL;DR

This paper studies the maximum number of colors in a strong vertex-monochromatic connection coloring of strong digraphs, especially line digraphs and tournaments, providing exact values and conditions.

## Contribution

It determines the exact maximum number of colors for SVMC-colorings in line digraphs and offers conditions for tournaments.

## Key findings

- Exact value of $smc_v(D)$ for line digraphs.
- Conditions for determining $smc_v(T)$ in tournaments.
- Advances understanding of vertex coloring in strong digraphs.

## Abstract

A vertex coloring of a strong digraph $D$ is a \emph{strong vertex-monochromatic connection coloring (SVMC-coloring)} if for every pair $u, v$ of vertices in $D$ there exists an $(u,v)$-path having all its internal vertices of the same color. Let $smc_v(D)$ denote the maximum number of colors used in an SVMC-coloring of a digraph $D$. In this paper we determine the value of $smc_v(D)$, whenever $D$ is the line digraph of a digraph. Also, if $T$ is a tournament, we give conditions to find the exact value of $smc_v(T)$.