  Finiteness conditions for $C^*$-algebras like AF-embeddability,
quasidiagonality, stable finiteness have been studied by many authors and shown
to be equivalent for certain classes of $C^*$-algebras. For example, Schfhauser
proves that these conditions are all equivalent for $C^*$-algebras of compact
topological graphs, and similar results were established by Clark, an Huef, and
Sims for $k$-graph algebras. If $C^*(E,\mathcal L)$ is a labeled graph
$C^*$-algebra over finite alphabet, it can be viewed as a $C^*$-algebra of a
compact topological graph. For these labeled graph $C^*$-algebras, we provide
conditions on labeled paths and show that they are equivalent to
AF-embeddability of $C^*(E,\mathcal L)$.
