Regular saturated graphs and sum-free sets

TL;DR

This paper extends the existence results of regular saturated graphs to larger complete graphs and odd cycles, using additive combinatorics techniques involving sum-free sets.

Contribution

It generalizes the existence of regular F-saturated graphs to K4, K5, and odd cycles using sum-free sets, and presents new partial results and open problems.

Findings

Existence of regular K3, K4, K5-saturated graphs for large n

Construction of regular C_{2k+1}-saturated graphs for infinitely many n

Open problem on sum-free sets related to saturated graphs

Abstract

In a recent paper, Gerbner, Patk\'{o}s, 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 $K_3$-saturated graph with $n$ vertices. We extend this result to both $K_4$ and $K_5$ and prove some partial results for larger complete graphs. Using a variation of sum-free sets from additive combinatorics, we prove that for all $k \geq 2$, there is a regular $C_{2k+1}$-saturated with $n$ vertices for infinitely many $n$. Studying the sum-free sets that give rise to $C_{2k+1}$-saturated graphs is an interesting problem on its own and we state an open problem in this direction.