4 problems

Let $k_1,k_2,\dots,k_n$ be positive integers. A subdivision of an oriented cycle $C(k_1,k_2,\dots,k_n)$ is obtained by replacing each arc by a directed path of length at least $1$,…

Let $C$ be an $n$-vertex oriented cycle, and let $D(n,p)$ be the binomial random digraph with $n$ vertices and edge probability $p$. Ferber–Long conjecture. If … then $D(n,p)$ cont…

Let $C$ be an oriented cycle, and let $\vec\chi(D)$ be the dichromatic number of a digraph $D$. Dichromatic-cycle conjecture. Every oriented cycle $C$ satisfies…

Let $C$ be an oriented cycle, and let $|V(C)|$ denote its order. Oriented-cycle conjecture. Every digraph $D$ with $\delta^+(D)\geq 2k-1$ contains a subdivision of any oriented cyc…