Saturation problems with regularity constraints

Abstract

For a graph F , we say that another graph G is F -saturated, if G is F -free and adding any edge to G would create a copy of F . We study for a given graph F and integer n whether there exists a regular n-vertex F -saturated graph, and if it does, what is the smallest number of edges of such a graph. We mainly focus on the case when F is a complete graph and prove for example that there exists a K 3 -saturated regular graph on n vertices for every large enough n.We also study two relaxed versions of the problem: when we only require that no regular F -free supergraph of G should exist or when we drop the F -free condition and only require that any newly added edge should create a new copy of F .

Generalized sum-free sets and cycle saturated regular graphs

“…This can be viewed as a regular version of saturation numbers much like the recently studied regular Turán numbers [4,5,11,17]. Gerbner et al [10] proved that rsat(n, K 3 ) exists for all sufficiently large n. This can also be proved using results of Haviv and Levy [14] on symmetric complete sum-free sets in Z n . Among several other interesting theorems, Gerbner et al obtained some partial results for K 4 .…”

“…Thus, we have a quadratic upper bound on rsat(n, C ℓ+1 ) for all n > 12ℓ 2 + 36ℓ + 24 for even ℓ ≥ 4. The lower bound rsat(n, C ℓ+1 ) = Ω(n 1+1/ℓ ), when this regular saturation number exists, is proved in [10]. As discussed in [19], the construction of Haviv and Levy implies rsat(n, C 3 ) = O(n 3/2 ), and so rsat(n,…”

“…A common theme in saturation is to add degree restrictions on the family of Fsaturated graphs; see [1,6,9,13] to name a few. A recent variant of sat(n, F ) was introduced by Gerbner, Patkós, Tuza, and Vizer [10]. Given a graph F and a positive integer n, one can ask if there exists an n-vertex F -saturated regular graph.…”

Regular saturated graphs and sum-free sets

“…Recently, Gerbner, Patkós, Tuza, and Vizer [10] studied the existence of regular F -saturated graphs and their number of edges. Let rsat(n, F ) be the minimum number of edges in an n-vertex regular F -saturated graph when such a graph exists.…”

“…Let rsat(n, F ) be the minimum number of edges in an n-vertex regular F -saturated graph when such a graph exists. It was noted in [10] that the principle concern is for which n and F does rsat(n, F ) exist. Gerbner et al [10] proved that rsat(n, K 3 ) exists for all n ≥ n 0 , and gave some partial results for rsat(n, K 4 ).…”