  Chen, Kitaev, M\"{u}tze, and Sun recently introduced the notion of universal
partial words, a generalization of universal words and de Bruijn sequences.
Universal partial words allow for a wild-card character $\diamond$, which is a
placeholder for any letter in the alphabet. We settle and strengthen
conjectures posed in the same paper where this notion was introduced. For
non-binary alphabets, we show that universal partial words have periodic
$\diamond$ structure and are cyclic, and we give number-theoretic conditions on
the existence of universal partial words. In addition, we provide an explicit
construction for a family of universal partial words over alphabets of even
size.
