  We apply Arveson's non-commutative boundary theory to dilate every
Toeplitz-Cuntz-Krieger family of a directed graph $G$ to a full Cuntz-Krieger
family for $G$. We do this by describing all representations of the Toeplitz
algebra $\mathcal{T}(G)$ that have unique extension when restricted to the
tensor algebra $\mathcal{T}_+(G)$. This yields an alternative proof to a result
of Katsoulis and Kribs that the $C^*$-envelope of $\mathcal T_+(G)$ is the
Cuntz-Krieger algebra $\mathcal O(G)$.
  We then generalize our dilation results further, to the context of colored
directed graphs, by investigating free products of operator algebras. These
generalizations rely on results of independent interest on complete injectivity
and a characterization of representations with the unique extension property
for free products of operator algebras.

Operator Algebras;Functional Analysis