Conflict-free vertex-connections of graphs

TL;DR

This paper studies the minimum number of colors needed to color vertices in a connected graph so that any two vertices are connected by a conflict-free path, with results varying based on the graph's connectivity.

Contribution

It provides new insights into the conflict-free vertex-connection number for different classes of graphs, especially trees and graphs with cut-vertices.

Findings

Easy characterization for 2-connected graphs

Complexity increases with the number of cut-vertices

Trees require a detailed analysis for conflict-free coloring

Abstract

A path in a vertex-colored graph is called \emph{conflict free} if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be \emph{conflict-free vertex-connected} if any two vertices of the graph are connected by a conflict-free path. This paper investigates the question: For a connected graph $G$, what is the smallest number of colors needed in a vertex-coloring of $G$ in order to make $G$ conflict-free vertex-connected. As a result, we get that the answer is easy for $2$-connected graphs, and very difficult for connected graphs with more cut-vertices, including trees.