A stability theorem for maximal K r+1 -free graphs

Abstract

For r ≥ 2, we show that every maximal K r+1 -free graph G on n vertices with (1 − 1 r ) n 2 2 − o(n r+1 r ) edges contains a complete r-partite subgraph on (1 − o(1))n vertices. We also show that this is best possible. This result answers a question of Tyomkyn and Uzzell.The classical theorem of Turán [12] tells us that, for an integer r ≥ 2, the maximum number of edges in a graph not containing a K r+1 is t r (n), and that T r (n) is the unique K r+1 -free graph attaining this maximum. Erdős and Simonovits [6,5,11] discovered that this extremal problem exhibits a certain 'stability' phenomenon: K r+1 -free graphs for which e(G) is close to t r (n) must resemble the Turán graph in an appropriate sense. In particular, they proved that every n-vertex, K r+1 -free graph with at least t r (n) − o(n 2 ) edges can be transformed into T r (n) by making at most o(n 2 ) edge deletions and additions.Beyond the seminal work of Erdős and Simonovits, we are lead to consider finer aspects of this phenomenon. More generally, it is natural to ask how the structure of a K r+1 -free graph G comes to resemble the Turán graph as the number of edges e(G) approaches the Turán number t r (n). For instance, Nikiforov and Rousseau [10], in the context of a Ramsey-theoretic problem, showed that for r ≥ 2 and ε sufficiently small (depending on r) the following holds:if G is an n-vertex K r+1 -free graph with e(G) ≥ 1 − 1 r − ε n 2 /2, then G contains an induced r-partite subgraph H with |H| ≥ (1 − 2ε 1/3 )n and δ(H) ≥ 1 − 1 r − 4ε 1/3 n. In other words, G must contain a large r-partite subgraph with minimum degree almost as large as δ(T r (n)).The interested reader should consult the survey of Nikiforov [9] for a few other stability results in a similar vein.

A Note on Stability for Maximal F-Free Graphs

“…This implies that exðn; FÞ 1 4 n 2 þ kn. We start by removing vertices of small degree, like the proofs of Theorem 1.1 in [15] and Theorem 1.2 in [21]. Let G 0 ¼ G and given G i on n i ¼ n À i vertices, if every vertex of G i has degree at least ð 1 2 À 1 11k Þn i , then we let G 0 ¼ G i .…”

“…Erd} os and Simonovits [10] extended the result of Andrásfai, Erd} os and Sós, while Simonovits [17] extended the result of Brouwer asymptotically to any ðr þ 1Þchromatic graph with a color-critical edge in place of K rþ1 . Using those results instead, the lemma below easily follows by the same proof as Lemma 2.3 in [15]. Lemma 1.5 Let r !…”

“…Does it contain a large complete r-partite subgraph? Popielarz, Sahasrabuddhe and Snyder [15] answered this question with the following theorem.…”

“…The first step of this proof holds for every graph in this generality, but we need a very delicate version that fully uses the property of having t r ðnÞ À oðn ðrþ1Þ=r Þ edges. Fortunately, the version due to Popielarz, Sahasrabuddhe and Snyder (Lemma 2.3 in [15]) easily extends to graphs with a color-critical edge. The version in [15] uses a result of Andrásfai, Erd} os and Sós [3] that determined the largest possible minimum degree in an n-vertex K rþ1 -free graph that is not r-partite, and a result of Brouwer [4] that determined the largest possible number of edges in such graphs.…”

New proofs of stability theorems on spectral graph problems

“…For a positive integer r ≥ 2, a graph G is said to be K r+1 -saturated (or maximal K r+1 -free) if it contains no copy of K r+1 , but the addition of any edge from the complement G creates at least one copy of K r+1 . In 2018, Popielarz, Sahasrabudhe and Snyder [43] proved the following stronger stability theorem for K r+1 -saturated graphs.…”