Generalized sum-free sets and cycle saturated regular graphs

TL;DR

This paper uses generalized sum-free sets to construct regular cycle-saturated graphs, establishing existence results for such graphs with specific cycle lengths and providing bounds and examples for sum-free sets.

Contribution

It introduces a novel method using sum-free sets to prove the existence of cycle-saturated regular graphs for various parameters.

Findings

Existence of n-vertex regular C_k-saturated graphs for all n ≥ n_k when k is odd and ≥ 5.

Construction of symmetric complete (ℓ,1)-sum-free sets in Z_n for even ℓ ≥ 4.

Identification of bounds and examples for minimal size of sum-free sets through computational search.

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.