Regular saturated graphs and sum-free sets

Abstract

In a recent paper, Gerbner, Patkós, Tuza and Vizer studied regular F-saturated graphs. One of the essential questions is given F, for which n does a regular n-vertex F-saturated graph exist. They proved that for all sufficiently large n, there is a regular K3-saturated graph with n vertices. We extend this result to both K4 and K5 and prove some partial results for larger complete graphs. Using a variation of sum-free sets from additive combinatorics, we prove that for all k≥2, there is a regular C2k+1-saturated with n vertices for infinitely many n. Studying the sum-free sets that give rise to C2k+1-saturated graphs is an interesting problem on its own and we state an open problem in this direction.