3 problems

For positive integers $k_1,k_2,\ldots,k_n$, let $C(k_1,k_2,\ldots,k_n)$ denote the oriented cycle formed by blocks of those lengths. A digraph is strongly connected if every ordere…

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…