Let $G$ be a finite group and $S$ be a subset of $G$. We define the Cayley graph of $G$ with respect to $S$ as follows, provided that $1 {\not\in} S$ and $S$ is inverse closed.
Definition: The Cayley graph of $G$ with respect to $S$, $Cay(G,S)$ is the graph whose vertices are the elements of $G$ and $g$ is adjacent to $gs$ for all $g \in G, \, s \in S$.
Suppose for a Cayley graph of a non-abelian group of odd order, say of order $p^2q$, where $p,q$ are distinct primes greater than 2, the generating set comprises of two elements $s,t$, where $|s|=p, |t|=q$. Then a cycle in the Cayley graph can be denoted by using the two elemnts $s,t$, as an example like
$stsst^{-1}s^{-1}...$ something like that.
If I count both $s$ and $s^{-1}$ terms as the number of terms of the $s$ form (say $n_1$) and if I count both $t$ and $t^{-1}$ terms as the number of terms of the $t$ form (say $n_2$) then in a case of considering a longest cycle which passes through all the vertices, $n_1 + n_2 = p^2q$.
In this situation can we say it has the maximum possible no of $s$ terms (the count including both $s,s^{-1}$) that can be present in a cycle and the maximum possible number of $t$ (the count including both $t,t^{-1}$) terms that can be present in a cycle?
Thanks a lot in advance.