Minimizing the number of 5-cycles in graphs with given edge-density

Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of cycles C5. We show that every graph of order n and size (1−1k)(n2), where k≥3 is an integer, contains at least (110−12k+1k2−1k3+25k4)n5+o(n5) copies of C5. This bound is optimal, since a matching upper bound is given by the balanced complete k-partite graph. The proof is based on the flag algebras framework. We also provide a stability result for 2≤k≤73

C5C_5 is almost a fractalizer

We determine the maximum number of induced copies of a 5-cycle in a graph on $n$ vertices for every $n$. Every extremal construction is a balanced iterated blow-up of the 5-cycle with the possible exception of the smallest level where for $n=8$, the M\"obius ladder achieves the same number of induced 5-cycles as the blow-up of a 5-cycle on 8 vertices. This result completes work of Balogh, Hu, Lidick\'y, and Pfender [Eur. J. Comb. 52 (2016)] who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method but we extend its use to small graphs.Comment: 24 page

Maximizing five-cycles in Kr-free graphs

The Erdos Pentagon problem asks to find an n-vertex triangle-free graph that is maximizing the number of 5-cycles. The problem was solved using flag algebras by Grzesik and independently by Hatami, Hladky, Kral, Norin, and Razborov. Recently, Palmer suggested the general problem of maximizing the number of 5-cycles in K_{k+1}-free graphs. Using flag algebras, we show that every K_{k+1}-free graph of order n contains at most 110k4(k4−5k3+10k2−10k+4)n5+o(n5) copies of C_5 for any k≥3, with the Turan graph begin the extremal graph for large enough n

Stability from graph symmetrisation arguments with applications to inducibility

We present a sufficient condition for the stability property of extremal graph problems that can be solved via Zykov's symmetrisation. Our criterion is stated in terms of an analytic limit version of the problem. We show that, for example, it applies to the inducibility problem for an arbitrary complete bipartite graph $B$, which asks for the maximum number of induced copies of $B$ in an $n$-vertex graph, and to the inducibility problem for $K_{2,1,1,1}$ and $K_{3,1,1}$, the only complete partite graphs on at most five vertices for which the problem was previously open.Comment: 41 page

Minimizing the number of 5-cycles in graphs with given edge-density

Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of cycles C5. We show that every graph of order n and size (1−1k)(n2), where k≥3 is an integer, contains at least (110−12k+1k2−1k3+25k4)n5+o(n5) copies of C5. This bound is optimal, since a matching upper bound is given by the balanced complete k-partite graph. The proof is based on the flag algebras framework. We also provide a stability result for 2≤k≤73.This is a manuscript made available through arxiv: https://arxiv.org/abs/1803.00165.</p