Conflict-Free Connection Number of Graphs with Four Bridges

Abstract

A path in an edge-colored graph G is called conflict-free if there is a color used on exactly one of its edges. An edge-colored graph G is said to be conflict-free connected if any two vertices are connected by a conflict-free path. The conflict-free connection number of a connected graph G, denoted by cfc(G), is defined as the smallest number of colors in order to make it conflict-free connected. Doan and Schiermeyer (Graphs Comb 37:1859–1871, 2021) conjectured that there exists an integer $n_0$, if $|E(G)|\ge {n-9\atopwithdelims ()2}+11$ for any connected graph G of order $n\ge n_0$ with 4 bridges, then $cfc(G)=2$ or G belongs to $\mathcal {G}_1\cup \mathcal {G}_2$, where $\mathcal {G}_1$ and $\mathcal {G}_2$ are two well-defined families of graphs. We disprove the conjecture. A refined version of the above conjecture is as follows: there exists an integer $n_0$, if $|E(G)|\ge {n-7\atopwithdelims ()2}+9$ for any connected graph G of order $n\ge n_0$ with four bridges, then $cfc(G)=2$ or G belongs to $\mathcal {G}_1\cup \mathcal {G}_2$. We prove that the conjecture is true if G has at least two nontrivial blocks.