Abstract
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.
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We thank the anonymous reviewer very much for several valuable comments and suggestions.
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Doan, T.D., Schiermeyer, I. Conflict-Free Connection Number and Size of Graphs. Graphs and Combinatorics 37, 1859–1871 (2021). https://doi.org/10.1007/s00373-021-02331-8
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DOI: https://doi.org/10.1007/s00373-021-02331-8