# Erd\"os-Gallai-type results for conflict-free connection of graphs

## TL;DR

This paper explores Erdős-Gallai-type conditions for the conflict-free connection number in graphs, establishing bounds and criteria for when graphs can be made conflict-free connected with a limited number of colors.

## Contribution

It introduces Erdős-Gallai-type theorems specifically for the conflict-free connection number, providing new theoretical bounds and conditions for graph coloring.

## Key findings

- Derived bounds for conflict-free connection numbers
- Established criteria for conflict-free connectivity
- Extended Erdős-Gallai results to conflict-free graph coloring

## Abstract

A path in an edge-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of its edges. An edge-colored graph is called \emph{conflict-free connected} if there is a conflict-free path between each pair of distinct vertices. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $\mathit{cfc}(G)$, is defined as the smallest number of colors that are required to make $G$ conflict-free connected. In this paper, we obtain Erd\"{o}s-Gallai-type results for the conflict-free connection numbers of graphs.