The k-forcing number of a graph G, denoted by $$F_k(G)$$ F k ( G ) , was introduced by Amos et al. It is a generalization of the zero forcing number of a graph G, denoted by Z(G). Amos et al. proved that for a connected graph G of order n with maximum degree $$\varDelta \ge 2$$ Δ ≥ 2 , $$Z(G)=F_1(G)\le \frac{(\varDelta -2)n+2}{\varDelta -1}$$ Z ( G ) = F 1 ( G ) ≤ ( Δ - 2 ) n + 2 Δ - 1 , and this inequality is sharp. Moreover, they posed a conjecture that $$Z(G)=F_1(G)=\frac{(\varDelta -2)n+2}{\varDelta -1}$$ Z ( G ) = F 1 ( G ) = ( Δ - 2 ) n + 2 Δ - 1 if and only if $$G=C_n$$ G = C n , $$G=K_{\varDelta +1}$$ G = K Δ + 1 or $$G=K_{\varDelta ,\varDelta }$$ G = K Δ , Δ . In this paper, we prove that this conjecture is true. Moreover, we point out a mistake in their paper and get a stronger result which shows that $$F_{n-1}(G)=1$$ F n - 1 ( G ) = 1 if and only if G is connected and $$F_k(G)=n-k$$ F k ( G ) = n - k if and only if $$G=K_n$$ G = K n for $$k\le n-2$$ k ≤ n - 2 .