Abstract
Gerbner, Patk\'{o}s, Tuza, and Vizer recently initiated the study of
$F$-saturated regular graphs. One of the essential problems in this line of
research is determining when such a graph exists. Using generalized sum-free
sets we prove that for any odd integer $k \geq 5$, there is an $n$-vertex
regular $C_k$-saturated graph for all $n \geq n_k$. Our proof is based on
constructing a special type of sum-free set in $\mathbb{Z}_n$. We prove that
for all even $\ell \geq 4$ and integers $n > 12 \ell^2 + 36 \ell + 24$, there
is a symmetric complete $( \ell , 1)$-sum-free set in $\mathbb{Z}_n$. We pose
the problem of finding the minimum size of such a set, and present some
examples found by a computer search.