The conflict-free connection number and the minimum degree-sum of graphs

In the context of an edge-coloured graph G, a path within the graph is deemed conflict-free when a colour is exclusively applied to one of its edges. The presence of a conflict-free path connecting any two unique vertices of an edge-coloured graph is what defines it as conflict-free connected. The conflict-free connection number, indicated by $cfc\left(G\right)$, is the fewest number of colours necessary to make G conflict-free connected. Consider the subgraph $C\left(G\right)$ of a connected graph G, which is constructed from the cut-edges of G. Let ${\sigma }_3\left(G\right)$ be the minimum degree-sum of any 3 independent vertices in G. In this study, we establish that for a connected graph G with an order of $n\ge 8$ and ${\sigma }_3\left(G\right)\ge n-1$, the following conditions hold: (1) $cfc\left(G\right)=3$ when $C\left(G\right)\cong K_{1,3}$; (2) $cfc\left(G\right)=2$ when $C\left(G\right)$ forms a linear forest. Moreover, we will now demonstrate that if G is a connected, non-complete graph with n vertices, where $n\ge 43$, $C\left(G\right)$ is a linear forest, $\delta \left(G\right)\ge 3$, and ${\sigma }_3\left(G\right)\ge \frac{3n-14}{5}$, then $cfc\left(G\right)=2$. Moreover, we also determine the upper bound of the number of cut-edges of a connected graph depending on the degree-sum of any three independent vertices.