  This paper investigates partitions which have neither parts nor hook lengths
divisible by $p$, referred to as $p$-core $p'$-partitions. We show that the
largest $p$-core $p'$-partition corresponds to the longest walk on a graph with
vertices $\{0, 1, \ldots, p-1\}$ and labelled edges defined via addition modulo
$p$. We also exhibit an explicit family of large $p$-core $p'$-partitions,
giving a lower bound on the size of the largest such partition which is of the
same degree as the upper bound found by McSpirit and Ono.
