Large $p$-core $p^{\mathrm{\prime }}$-partitions and walks on the additive residue graph

This paper investigates partitions which have neither parts nor hook lengths divisible by $p$, referred to as $p$-core $p^{\mathrm{\prime }}$-partitions. We show that the largest $p$-core $p^{\mathrm{\prime }}$-partition corresponds to the longest walk on a graph with vertices $\{0,1,\dots ,p-1\}$ and labelled edges defined via addition modulo $p$. We also exhibit an explicit family of large $p$-core $p^{\mathrm{\prime }}$-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.