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(G), is the fewest number of colours necessary to make G conflict-free connected. Consider the subgraph C(G) of a connected graph G, which is constructed from the cut-edges of G. Let σ3(G) 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≥8 and σ3(G)≥n−1, the following conditions hold: (1) cfc(G)=3 when C(G)≅K1,3; (2) cfc(G)=2 when C(G) forms a linear forest. Moreover, we will now demonstrate that if G is a connected, non-complete graph with n vertices, where n≥43, C(G) is a linear forest, δ(G)≥3, and σ3(G)≥3n−145, then cfc(G)=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.