Stability results for graphs with a critical edge

Abstract

The classical stability theorem of Erdős and Simonovits states that, for any fixed graph with chromatic number k + 1 ≥ 3, the following holds: every n-vertex graph that is H-free and has within o(n 2 ) of the maximal possible number of edges can be made into the k-partite Turán graph by adding and deleting o(n 2 ) edges. In this paper, we prove sharper quantitative results for graphs H with a critical edge, both for the Erdős-Simonovits Theorem (distance to the Turán graph) and for the closely related question of how close an H-free graph is to being k-partite. In many cases, these results are optimal to within a constant factor.

Strong forms of stability from flag algebra calculations

“…As far as we know, the above results in [15,41] and some recent work of Norin and Yepremyan [33,34] (who considered the Turán problem for hypergraphs) are the only known examples where perfect stability was established for a non-trivial problem. Furthermore, almost all proofs where the classical stability and the exact result were derived from a flag algebra computation were rather ad-hoc and taylored to a particular problem.…”

“…By (43) and recalling that we are only interested in points that belong to the interior of D, we get that x 3 = 1 − x 1 − x 2 − x 3 , while by subtracting (41) and (42), we get that x 1 = x 2 . Combining these two we get, in particular, that x 1 +x 3 = 1/2.…”

“…The perfect (and thus also robust) stability in the case when F consists of a clique K t follows from results of Füredi [15] (who considered the distance to being (t − 1)-partite instead of complete (t − 1)-partite as we do in this paper). Very recently, Roberts and Scott [41] improved on Füredi's result by extending it to all colour critical graphs and giving a sharper bound on the distance.…”

Making Kr+1-free graphs r-partite

“…In [11] Füredi's result was strengthened for some values of r. Roberts and Scott [15] showed that D r (G) = O(t 3/2 /n) when t δn 2 , and that this result is sharp up to a constant factor. They also proved a more general result for H-free graphs where H is an edge-critical graph.…”

A Note on Stability for Maximal F-Free Graphs

“…K r and r-chromatic graphs with a color-critical edge often behave similarly in extremal questions, see e.g. [17] and see [16] for several stability results. Another reason to assume that this conjecture might hold, and it might be easier to prove this than Conjecture 1.3 is the following.…”