  The giant $k$-core --- maximal connected subgraph of a network where each
node has at least $k$ neighbors --- is important in the study of phase
transitions and in applications of network theory. Unlike Erd\H{o}s-R\'enyi
graphs and other random networks where $k$-cores emerge discontinuously for
$k\ge 3$, we show that transitive linking (or triadic closure) leads to 3-cores
emerging through single or double phase transitions of both discontinuous and
continuous nature. We also develop a $k$-core calculation that includes
clustering and provides insights into how high-level connectivity emerges.
