The maximum number of cliques in dense graphs

AbstractDenote the number of vertices of G by |G|. A clique of graph G is a maximal complete subgraph. The density ω(G) is the number of vertices in the largest clique of G. If ω(G)⩾12|G|, then G has at most 2|G|−ω(G) cliques. The extremal graphs are then examined as well

On the generalized Tur\'an problem for odd cycles

In 1984, Erd\H{o}s conjectured that the number of pentagons in any triangle-free graph on $n$ vertices is at most $(n/5)^5$, which is sharp by the balanced blow-up of a pentagon. This was proved by Grzesik, and independently by Hatami, Hladk\'y, Kr\'al', Norine and Razborov. As an extension of this result for longer cycles, we prove that for each odd $k\geq 7$, the balanced blow-up of $C_k$ (uniquely) maximises the number of $k$-cycles among $C_{k-2}$-free graphs on $n$ vertices, as long as $n$ is sufficiently large. We also show that this is no longer true if $n$ is not assumed to be sufficiently large. Our result strengthens results of Grzesik and Kielak who proved that for each odd $k\geq 7$, the balanced blow-up of $C_k$ maximises the number of $k$-cycles among graphs with a given number of vertices and no odd cycles of length less than $k$. We further show that if $k$ and $\ell$ are odd and $k$ is sufficiently large compared to $\ell$, then the balanced blow-up of $C_{\ell+2}$ does not asymptotically maximise the number of $k$-cycles among $C_{\ell}$-free graphs on $n$ vertices. This disproves a conjecture of Grzesik and Kielak.Comment: 15 page

Many triangles with few edges

Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with n vertices and maximum degree at most r, where n = a(r + 1) + b and 0 ≤ b ≤ r, aKr+1 ∪ Kb has the maximum number of complete subgraphs, answering a question of Galvin. Gan, Loh and Sudakov conjectured that aKr+1 ∪Kb also maximizes the number of complete subgraphs Kt for each fixed size t ≥3, and proved this for a = 1. Cutler and Radcliffe proved this conjecture for r ≤ 6. We investigate a variant of this problem where we fix the number of edges instead of the number of vertices. We prove that aKr+1 ∪C(b), where C(b) is the colex graph on b edges, maximizes the number of triangles among graphs with m edges and any fixed maximum degree r ≤ 8, where m = a(r+1 2 ) + b and 0 ≤ b < (r+1 2 ). Mathematics Subject Classifications: 05