7 problems

In a digraph $D=(V,A)$, a dicut is a set of arcs of the form $\delta^+(U)$ for a nonempty proper subset $U\subsetneqq V$ such that $\delta^-(U)=\emptyset$, and a dijoin is a set of…

Let $D=(V,A)$ be a digraph whose underlying undirected graph is $5$-chordal, meaning that it has no chordless cycle of length more than $5$. A dicut is a set of arcs directed acros…

Dijoin weight decomposition conjecture. There exist weighted digraphs $(D,c)$ and $(D,c')$ such that

Dijoin-clutter $\tau=2$ conjecture. The weighted digraph $(D,w)$ has a dicut of weight two.

Barnette-type conjecture. There exists a rounded $1$-factor $J$ such that

Connected-support conjecture. The statement $[\operatorname{wt},\tau]$ should hold for every such weighted digraph.

Guenin's conjecture. The statement $[\operatorname{wt},\tau]$ should hold when $D$ has a super source and a super sink.