Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle

Abstract

Let C(n) denote the maximum number of induced copies of 5-cycles in graphs on n vertices. For n large enough, we show thatand a, b, c, d, e are as equal as possible.Moreover, if n is a power of 5, we show that the unique graph on n vertices maximizing the number of induced 5-cycles is an iterated blow-up of a 5-cycle.The proof uses flag algebra computations and stability methods.

Closing in on Hill's Conjecture

“…Our proof makes essential use of flag algebras. This powerful tool, introduced by Razborov [38], has been the basis of several recent groundbreaking results in a variety of combinatorial and geometric problems, such as [10,12,13,19,25,27,37,39], to name just a few.…”

Inducibility of d-ary trees

“…The concept of inducibility is still gaining consideration from several research groups; see [13] and [9] for some recent results on blow-up of graphs and graphs on four vertices, respectively. The language of flag algebra was also employed recently in [1] to derive the inducibility of the cycle on five vertices, thereby settling a particular case of a conjecture formulated in [16].…”

Strong forms of stability from flag algebra calculations

“…Under the above conventions, consider the problem of maximising Λ(G) over admissible graphs G of given order n. Namely, we define the extremal function Λ(n, G) := max{Λ(G) : G ∈ G, v(G) = n} (2) and its density version λ(n, G) := Λ(n, G)/ n κ . It is not hard to show (see Lemma 2.2) that the sequence (λ(n, G)) ∞ n=κ is non-increasing and therefore the following limit exists:…”

“…For a family H of graphs we define ∆ edit (G, H) := min{∆ edit (G, H) : H ∈ H 0 n } and δ edit (G, H) := min{δ edit (G, H) : H ∈ H 0 n }. We say that our problem (2) is robustly B-stable (resp. perfectly B-stable) if there is C > 0 such that for every graph G ∈ G of order n C we have δ edit (G, B()) C max (1/n, λ(n, G) − λ(G)) , (resp.…”