An edge-coloured graph G is called conflict-free connected if every two distinct vertices are connected by at least one path, which contains a colour used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is the smallest number of colours needed in order to make it conflict-free connected. For a graph G, let C(G) be the subgraph of G induced by its set of bridges. Our main results are the following: (1) Let \(k\ge 2\) and G be a connected graph of order n containing bridges. If \(\vert E(G)\vert \ge {{n-2k}\atopwithdelims ()2}+2k+1\), then \(cfc(G)\le k\) or \(\varDelta (C(G)) \ge k+1.\) (2) Let G be a connected graph of order n with \(t \ge 4\) bridges such that \(n \ge t+15.\) If \(\vert E(G)\vert \ge {{n-t-2}\atopwithdelims ()2}+t+4\), then \(cfc(G)=2\) or G belongs to a class of exceptional graphs.

如果每两个不同的顶点通过至少一条路径连接，则该边缘着色的图形G称为无冲突连接，该路径至少包含一条在其边缘之一上使用的颜色。在无冲突的连接数的连通图的G ^，记为CFC（ģ），是的，以便在需要的颜色，使其无冲突连接的最小数目。对于图G，令C（G）是由其桥集合引起的G的子图。我们的主要结果如下：（1）令\（k \ ge 2 \）和G为n阶连通图包含桥梁。如果\（\ vert E（G）\ vert \ ge {{n-2k} \ atopwithdelims（）2} + 2k + 1 \），则\（cfc（G）\ le k \）或\（\ varDelta（ C（G））\ ge k + 1。\）（2）令G是具有\（t \ ge 4 \）桥的阶n的连通图，使得\（n \ ge t + 15。\）如果\ （\ vert E（G）\ vert \ ge {{nt-2} \ atopwithdelims（）2} + t + 4 \），则\（cfc（G）= 2 \）或G属于一类特殊图形。