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.
Categories: math.CO