# Conflict-free (vertex)-connection numbers of graphs with small diameters

## TL;DR

This paper determines the exact conflict-free (vertex-)connection numbers for graphs with small diameters, specifically for trees and graphs with diameters up to 4, advancing understanding of conflict-free connectivity in graph theory.

## Contribution

The paper provides exact values of conflict-free (vertex-)connection numbers for trees with diameters 2-4 and extends results to all graphs with diameter at most 4, except some specific cases.

## Key findings

- Exact conflict-free connection number for trees with diameters 2, 3, 4
- Conflict-free connection number for graphs with diameter ≤ 4
- Exact conflict-free vertex-connection number for graphs with diameter ≤ 4

## Abstract

A path in an(a) edge(vertex)-colored graph is called a conflict-free path if there exists a color used on only one of its edges(vertices). An(A) edge(vertex)-colored graph is called conflict-free (vertex-)connected if for each pair of distinct vertices, there is a conflict-free path connecting them. For a connected graph $G$, the conflict-free (vertex-)connection number of $G$, denoted by $cfc(G)(\text{or}~vcfc(G))$, is defined as the smallest number of colors that are required to make $G$ conflict-free (vertex-)connected. In this paper, we first give the exact value $cfc(T)$ for any tree $T$ with diameters $2,3$ and $4$. Based on this result, the conflict-free connection number is determined for any graph $G$ with $diam(G)\leq 4$ except for those graphs $G$ with diameter $4$ and $h(G)=2$. In this case, we give some graphs with conflict-free connection number $2$ and $3$, respectively. For the conflict-free vertex-connection number, the exact value $vcfc(G)$ is determined for any graph $G$ with $diam(G)\leq 4$.