I have two circulant Cayley digraphs: that is, Cayley digraphs *X* = Cay(ℤ/*m*, *S*) and *Y* = Cay(ℤ/*n*, *T*), for odd integers *m* < *n*, and sets with sizes |*S*| = (*m* − 1)/2, and |*T*| = (*n* − 1)/2.


These digraphs are antisymmetric, in that *S* is disjoint from −*S*, and *T* is disjoint from −*T*. (It follows that for each distinct pair of vertices *a,b* in either graph, there is either an arc from *a* to *b*, or vice versa.)


**Question.** What conditions on *m*, *n*, *S*, and *T* must hold for *X* to be an induced directed subgraph of *Y*?



Conditions for subgraph relationship in circulant Cayley digraphs 
I very much doubt that there is a nice answer for this. 
I suspect that this question
is not essentially easier than the more general problem, where we allow $X$ to be any
tournament. If $n$ is a prime congruent to 3 mod 4 and $T$ is the set of non-zero
squares in $\mathbb{Z}/n$, the Cayley graph $Y$ is the Paley tournament. It follows from an
old result of Graham and Spencer that any smallish tournament is an induced directed
subgraph of $Y$. (Here ``smallish'' is technical term that means something like $\log(n)$,
or perhaps $\sqrt{(\log(n))}$.)


