  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.
