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It is known that graph braid groups have a close relationships with right-angled Artin groups, and in particular, Crist and Wiest have shown that all graph braid groups embed into some right-angled Artin group. In that same paper, they also show that most right-angled Artin groups embed into some classical (pure) braid group, and here we will describe an explicit construction to extend the result to all right-angled Artin groups. Thus as an immediate consequence, all graph braid groups embed into some classical (pure) braid group. Also if time permits, we will demonstrate that all possible homomorphisms for the $n$-strand classical braid group for $n > 4$ to an arbitrary right-angled Artin group will have an image isomorphic to $\mathbb{Z}$.