On the Number of Pentagons in Triangle-Free Graphs

Using the formalism of flag algebras, we prove that every triangle-free graph $G$ with $n$ vertices contains at most $(n/5)^5$ cycles of length five. Moreover, the equality is attained only when $n$ is divisible by five and $G$ is the balanced blow-up of the pentagon. We also compute the maximal number of pentagons and characterize extremal graphs in the non-divisible case provided $n$ is sufficiently large. This settles a conjecture made by Erd\H{o}s in 1984.Comment: 16 pages, accepted to Journal of Combinatorial Theory Ser.

Pentagons in triangle-free graphs

For all $n\ge 9$, we show that the only triangle-free graphs on $n$ vertices maximizing the number $5$-cycles are balanced blow-ups of a 5-cycle. This completely resolves a conjecture by Erd\H{o}s, and extends results by Grzesik and Hatami, Hladk\'y, Kr\'{a}l', Norin and Razborov, where they independently showed this same result for large $n$ and for all $n$ divisible by $5$.Comment: 6 page

Non-three-colorable common graphs exist

A graph H is called common if the total number of copies of H in every graph and its complement asymptotically minimizes for random graphs. A former conjecture of Burr and Rosta, extending a conjecture of Erdos asserted that every graph is common. Thomason disproved both conjectures by showing that the complete graph of order four is not common. It is now known that in fact the common graphs are very rare. Answering a question of Sidorenko and of Jagger, Stovicek and Thomason from 1996 we show that the 5-wheel is common. This provides the first example of a common graph that is not three-colorable.Comment: 9 page

Many HH-copies in graphs with a forbidden tree

For graphs $H$ and $F$, let $\operatorname{ex}(n, H, F)$ be the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices. The study of this function, which generalises the well-studied Tur\'an numbers of graphs, was initiated recently by Alon and Shikhelman. We show that if $F$ is a tree then $\operatorname{ex}(n, H, F) = \Theta(n^r)$ for some integer $r = r(H, F)$, thus answering one of their questions.Comment: 9 pages, 1 figur

A note on the maximum number of triangles in a C5-free graph

We prove that the maximum number of triangles in a C5-free graph on n vertices is at most [Formula presented](1+o(1))n3/2, improving an estimate of Alon and Shikhelman [Alon, N. and C. Shikhelman, Many T copies in H-free graphs. Journal of Combinatorial Theory, Series B 121 (2016) 146-172]. © 2017 Elsevier B.V

On the maximum number of five-cycles in a triangle-free graph

Using Razborov's flag algebras we show that a triangle-free graph on n vertices contains at most (n/5)^5 cycles of length five. It settles in the affirmative a conjecture of Erdos.Comment: After minor revisions; to appear in JCT