# Strong conflict-free connection of graphs

## TL;DR

This paper introduces the concept of strong conflict-free connection in graphs, providing bounds on the number of colors needed and characterizations for specific graph classes, including cubic graphs.

## Contribution

It establishes bounds on the strong conflict-free connection number for graphs with triangles and characterizes graphs with specific strong conflict-free connection numbers.

## Key findings

- For graphs with t triangles, scfc(G) ≤ m - 2t.
- Characterization of graphs with scfc(G) in {1, m-3, m-2, m-1, m}.
- Complete characterization of cubic graphs with scfc(G)=2.

## Abstract

A path $P$ in an edge-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of the edges of $P$. An edge-colored graph $G$ is called \emph{conflict-free connected} if for each pair of distinct vertices of $G$ there is a conflict-free path in $G$ connecting them. The graph $G$ is called \emph{strongly conflict-free connected }if for every pair of vertices $u$ and $v$ of $G$ there exists a conflict-free path of length $d_G(u,v)$ in $G$ connecting them. For a connected graph $G$, the \emph{strong conflict-free connection number} of $G$, denoted by $\mathit{scfc}(G)$, is defined as the smallest number of colors that are required in order to make $G$ strongly conflict-free connected. In this paper, we first show that if $G_t$ is a connected graph with $m$ $(m\geq 2)$ edges and $t$ edge-disjoint triangles, then $\mathit{scfc}(G_t)\leq m-2t$, and the equality holds if and only if $G_t\cong S_{m,t}$. Then we characterize the graphs $G$ with $scfc(G)=k$ for $k\in \{1,m-3,m-2,m-1,m\}$. In the end, we present a complete characterization for the cubic graphs $G$ with $scfc(G)=2$.