On the $p^{\mathrm{\prime }}$-subgraph of the Young graph

Let $p$ be a prime number. In this article we study the restriction to ${\mathfrak{S}}_{n-1}$ of irreducible characters of degree coprime to $p$ of ${\mathfrak{S}}_n$. In particular, we study the combinatorial properties of the subgraph ${\mathbb{Y}}_{p^{\mathrm{\prime }}}$ of the Young graph $\mathbb{Y}$. This is an extension to odd primes of the work done by Ayyer, Prasad and Spallone for $p=2$.