13 problems

Let $G$ be a digraph, let $\beta(G)$ denote the minimum size of a feedback arc set, and let $\gamma(G)$ denote the number of non-adjacent unordered pairs of vertices of $G$. An $r$…

Half-gap conjecture. If $G$ is a $3$-free digraph, then

Pointwise extremal conjecture. Equality holds for all $q\in(0,1)$:

Let ${\rm fasd}(\Delta,g)$ denote the supremum, over the relevant weighted orgraphs of maximum degree at most $\Delta$ and directed girth at least $g$, of the normalized feedback a…

Let $D$ be a non-acyclic planar digraph. Write $g(D)$ for its directed girth, the minimum length of a directed cycle, and let ${\rm fasd}(D)$ be the maximum number of pairwise arc-…

Hanauer et al.'s conjecture. If $Delta\leq 5$, then

Hanauer's conjecture. If $Delta\leq 5$, then

Let $H$ be a tournament. It has the strong Erdős–Hajnal property if there exists $epsilon_H>0$ such that every $H$-free tournament $T$ contains disjoint vertex subsets $A$ and $B$…

Let $G=(V,E)$ be a digraph on $n$ vertices, let $\beta(G)$ denote the minimum size of a feedback arc set, and let an $(r-1)$-free digraph be one containing no directed cycle of len…

Let $r$ be an even integer with $r\geq 4$, and let an $r$-free digraph be a digraph containing no directed cycle of length at most $r$. For a digraph $G$, let $\beta(G)$ be the min…

Woodall's conjecture. There exists a packing of $g$ disjoint feedback arc sets in $D$.

Let $H$ be an oriented graph, and write $\operatorname{FAS}(H)$ for the minimum feedback arc set size of $H$. In the biased orientation game, Maker wins when the resulting tourname…

Let $H$ be a fixed graph, and consider the biased orientation game on a complete graph in which Maker wins when the resulting tournament contains a copy of $H$. The minimum feedbac…