Large $$p$$-Core $$p'$$-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 -core $$p'$$ -partitions. We show that the largest -partition corresponds longest walk on a graph with vertices $$\{0, 1, \ldots p-1\}$$ and labelled edges defined via addition modulo . also exhibit an explicit family of large -partitions, giving lower bound size such partition is same degree upper found McSpirit Ono.