On the maximum number of copies of H in graphs with given size and order

Abstract

We study the maximum number ex(n, e, H) of copies of a graph H in graphs with given number of vertices and edges. We show that for any fixed graph H, ex(n, e, H) is asymptotically realized by the quasi-clique provided that the edge density is sufficiently large. We also investigate a variant of this problem, when the host graph is bipartite.

The upper tail problem for induced 4‐cycles in sparse random graphs

“…They solved this problem for some class of graphs which includes all complete bipartite graphs K s,t . Here we prove a similar statement to the ones in [7,19].…”

“…This follows by a standard double counting argument and as the set of all vertices of degree at most R is an independent set. We thus obtain (19) as…”

“…The analog of this construction in our case is a balanced and complete bipartite graph with $$$ \Theta (np) $$$ vertices. Surprisingly, although amongst all graphs with a given number of edges the complete and balanced bipartite graph maximizes the number of induced copies of $$$ {C}_4 $$$ (see [7, 19] and Lemma 6.2) planting it does not always give the strongest lower bound on the upper tail probability.…”

Regular Turán numbers and some Gan-Loh-Sudakov-type problems

“…In this section, we focus on the question posed in [14] of maximizing the number k t (G) of cliques K t in G given both the order and size as well as the maximum degree. Up to the maximum degree condition, this question appeared for example also in [9]. Assuming the conjecture in [14] is true, the extremal graph with n = a(r + 1) + b (here b ≤ r) and m ≤ a r+1 2 + b 2 would be the union of K r+1 s and a colex graph.…”

Regular Turán numbers and some Gan–Loh–Sudakov‐type problems

“…They also wondered about the problem of maximizing $${k}_{t}(G)$$ in $$G$$ given the order $$n$$ and size $$m$$, as well as the maximum degree of $$G$$. (Up to the maximum degree condition, this question appeared, e.g., also in [11]. ) As will become apparent, the most interesting case here is that of regular graphs.…”