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

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