Related papers: On the maximum number of five-cycles in a triangle…

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…

One of Erdos's conjectures states that every triangle-free graph on $n$ vertices has an induced subgraph on $n/2$ vertices with at most $n^2/50$ edges. We report several partial results towards this conjecture. In particular, we establish…

We introduce a new approach and prove that the maximum number of triangles in a $C_5$-free graph on $n$ vertices is at most $$(1 + o(1)) \frac{1}{3 \sqrt 2} n^{3/2}.$$ We also show a connection to $r$-uniform hypergraphs without (Berge)…

The Erd\H{o}s 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, Hladk\'{y}, Kr\'{a}l', Norin, and…

In 1975, Erd\H{o}s asked the following natural question: What is the maximum number of edges that an $n$-vertex graph can have without containing a cycle with all diagonals? Erd\H{o}s observed that the upper bound $O(n^{5/3})$ holds since…

Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle $C_5$. We show that every graph of order $n$ and size $\left( 1-\frac{1}{k}\right)\binom{n}{2}$, where $k\ge 3$…

We prove that the maximum number of triangles in a $C_5$-free graph on $n$ vertices is at most $\frac{1}{2 \sqrt 2} (1 + o(1)) n^{3/2}$, improving an estimate of Alon and Shikhelman.

It is shown that for $n\geq 141$, among all triangle-free graphs on $n$ vertices, the complete equibipartite graph is the unique triangle-free graph with the greatest number of cycles.

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,…

We prove that for each odd integer $k \geq 7$, every graph on $n$ vertices without odd cycles of length less than $k$ contains at most $(n/k)^k$ cycles of length $k$. This generalizes the previous results on the maximum number of pentagons…

We conjecture that the balanced complete bipartite graph $K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil}$ contains more cycles than any other $n$-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent…

Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the…

Fix $k \ge 2$ and let $H$ be a graph with $\chi(H) = k+1$ containing a critical edge. We show that for sufficiently large $n$, the unique $n$-vertex $H$-free graph containing the maximum number of cycles is $T_k(n)$. This resolves both a…

More than twenty years ago Erd\H{o}s conjectured~\cite{E1} that a triangle-free graph $G$ of chromatic number $k \geq k_0(\varepsilon)$ contains cycles of at least $k^{2 - \varepsilon}$ different lengths as $k \rightarrow \infty$. In this…

Bollob\'as proved that for every $k$ and $\ell$ such that $k\mathbb{Z}+\ell$ contains an even number, an $n$-vertex graph containing no cycle of length $\ell \bmod k$ can contain at most a linear number of edges. The precise (or asymptotic)…

The celebrated Mantel's theorem states that any triangle-free graph on $n$ vertices contains at most $\left\lfloor n^2/4\right\rfloor$ edges. It is natural to ask how many triangles must exist in a graph with more than $\left\lfloor…

Finding the maximum number of induced cycles of length $k$ in a graph on $n$ vertices has been one of the most intriguing open problems of Extremal Graph Theory. Recently Balogh, Hu, Lidick\'{y} and Pfender answered the question in the case…

We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications include: * Every $n$-vertex graph with no 5-cycle can be made…

Hajos' conjecture that every simple even graph on $n$ vertices can be decomposed into at most $(n-1)/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp.…

We show that every $r$-uniform hypergraph on $n$ vertices which does not contain a tight cycle has at most $O(n^{r-1} (\log n)^5)$ edges. This is an improvement on the previously best-known bound, of $n^{r-1} e^{O(\sqrt{\log n})}$, due to…