An edge-colored graph G is conflict-free connected if any two of its vertices are connected by a path, which contains a color used on exactly one of * the minimum degree δ(G) ≥ max{3, n−4 5 }, then cf c(G) = 2. The bound on the minimum degree is best possible. Next, we prove that, if G is a connected non-complete graph of order n ≥ 33 with C(G) being a linear forest and with d(x) + d(y) ≥ 2n−9 5 for each pair of two nonadjacent vertices x, y of V (G), then cf c(G) = 2. Both bounds, on the order n and the degree sum, are tight. Moreover, we prove several results concerning relations between degree conditions on G and the number of cut edges in G.

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