9 problems

Let $D$ be a digraph. A branching is a digraph whose components are arborescences, and write $\gamma(D)$ for the directed fractional packing parameter and $\Delta^-$ and $\Delta^+$…

Bang-Jensen and Wang's conjecture. Every $2$-arc-strong semicomplete split digraph $D$ contains a good $(u,v)$-pair for every choice of vertices $u,v$ in $D$.

Thomassen's conjecture. There exists an integer $K$ such that every $K$-arc-strong digraph $D=(V,A)$ has a good $(u,v)$-pair for every choice of $u,v\in V$.

Bang-Jensen and Gutin's conjecture. There exists a polynomial algorithm for deciding, given a quasi-transitive digraph $D=(V,A)$ and two vertices $u,v$ of $V$, whether $D$ has a go…

Let $D$ be a digraph on at least $L$ vertices. Write $\lambda(D)$ for its arc-connectivity and $\alpha(D)$ for its independence number, and let $B_s^+$ denote an out-branching root…

Let $D$ be a digraph on at least $K$ vertices. Write $\lambda(D)$ for its arc-connectivity and $\alpha(D)$ for its independence number. A non-separating out-branching is an out-bra…

Let $D=(V,A)$ be a digraph. Write $\alpha(D)$ for its independence number, let $\lambda(D)$ denote its arc-connectivity, and let $B_s^+$ and $B_s^-$ denote, respectively, an out-br…

Prescribed-roots conjecture. Every $3$-arc-strong digraph $D=(V,A)$ with $calpha(D)=2$ has a pair of arc-disjoint branchings $B_s^+,B_t^-$ for every choice of $s,t\ro V$.

Let $D=(V,A)$ be a digraph, let $\kappa$ be an infinite cardinal, and let $\mathcal{B}_i=(V_i,A_i)$ for $i<\kappa$ be edge-disjoint branchings in $D$. Define … Suppose that…