10 problems

Let $G$ be a triangle-free graph on $n$ vertices, and let $N(G;C_5)$ denote the number of five-cycles in $G$. Erdős's conjecture. A triangle-free graph on $n$ vertices has at most…

Erdős's conjecture. The maximum number of cycles of length $5$ in $G$ is

Let $ex(n;C_{k+1};k)$ denote the maximum number of directed copies of the cycle $C_{k+1}$ in a $k$-geodetic digraph of order $n$. The asymptotic cycle-count conjecture asserts that…

Chakraborti–Loh uniqueness conjecture. The graph $K_{s-2}+\overline{K_{n-s+2}}$ is the unique $n$-vertex $K_s$-saturated graph minimizing the number of copies of $C_r$. This is a p…

Unique extremal graph conjecture. The graph $K_{s-2}\ast\overline{K}_{n-s+2}$ is the unique graph minimizing the number of cycles of length $r$ among all $n$-vertex $K_s$-saturated…

An induced $k$-cycle is a cycle of length $k$ whose vertices span no additional edges. An iterated blow-up of $C_k$ is obtained by repeatedly replacing vertices by independent sets…

A graph is triangle-free if it contains no cycle of length $3$. For a graph on $n$ vertices, a cycle of length $5$ is a pentagon, and the balanced blow-up of $C_5$ is obtained by r…

Independence conjecture. For $b_r$-regular permutations, any two cycle indicators are independent if and only if they are $r-1$-separated.

Let $G$ be a triangle-free graph on $n$ vertices, and let a cycle of length $5$ mean a cycle consisting of five edges. Erdős's conjecture. The maximum number of cycles of length…

Minimum-cycle conjecture. If $\chi(G)>k$, then $G$ contains at least