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An edge-coloured graph G is called properly connected if every two vertices are connected by a proper path. The proper connection number of a connected graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. Susan A. van Aardt et al. gave a sufficient condition for the proper connection number tobe at most k in terms of the size of graphs. In this note, our main result is the following, by adding a minimum degree condition: Let G be a connected graph of order n, k ≥ 3. IfFurthermore, if k = 2 and δ = 2, pc(G) ≤ 2, except G ∈ {G 1 , G n } (n ≥ 8), where G 1 = K 1 ∨ 3K 2 and G n is obtained by taking a complete graph K n−5 and K 1 ∨ (2K 2 ) with an arbitrary vertex of K n−5 and a vertex with d(v) = 4 in K 1 ∨ (2K 2 ) being joined. If k = 2, δ ≥ 3, we conjecture pc(G) ≤ 2, where m takes the value 1 if δ = 3 and 0 if δ ≥ 4 in the assumption.

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