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An edge-colored graph $G$ is \emph{conflict-free connected} if, between eachpair of distinct vertices, there exists a path containing a color used onexactly one of its edges. The \emph{conflict-free connection number} of aconnected graph $G$, denoted by $cfc(G)$, is defined as the smallest number ofcolors that are needed in order to make $G$ conflict-free connected. In thispaper, we determine all trees $T$ of order $n$ for which $cfc(T)=n-t$, where$t\geq 1$ and $n\geq 2t+2 $. Then we prove that $1\leq cfc(G)\leq n-1$ for aconnected graph $G$, and characterize the graphs $G$ with$cfc(G)=1,n-4,n-3,n-2,n-1$, respectively. Finally, we get theNordhaus-Gaddum-type theorem for the conflict-free connection number of graphs,and prove that if $G$ and $\overline{G}$ are connected, then $4\leqcfc(G)+cfc(\overline{G})\leq n$ and $4\leq cfc(G)\cdotcfc(\overline{G})\leq2(n-2)$, and moreover, $cfc(G)+cfc(\overline{G})=n$ or$cfc(G)\cdot cfc(\overline{G})=2(n-2)$ if and only if one of $G$ and$\overline{G}$ is a tree with maximum degree $n-2$ or a $P_5$, and the lowerbounds are sharp.

Nordhaus-Gaddum-type theorem for conflict-free connection number of graphs