In this paper, we obtain sharp Faber-Krahn inequalities for the first Dirichlet eigenvalue of the combinatorial $p$-Laplacian on connected graphs with a fixed number of vertices or with a fixed number of edges. More precisely, we show that the minimum of the first $p$-Dirichlet eigenvalues of connected graphs with boundary that consist of $n$ vertices or $n$ edges is achieved only on the tadpole graph $T_{n,3}$ when $p>1$.