4 problems

Let $d\ge 2$ be fixed. A separator of a graph on $n$ vertices is a partition of its vertex set into sets $A,B,C$ with $cn\le |A|\le |B|\le (1-c)n$, for a fixed constant $c$ satisfy…

Let $P$ range over all $3$-dimensional polytopes with $n$ vertices, and let $f$ range over all generic linear functionals on $P$. Write $G(P,f)$ for the flip graph associated with…

Let $P$ be a simple $3$-dimensional polytope with $2n$ vertices, and let $f$ be a generic linear functional on $P$. Let $\b5(P,f)$ denote the number of $f$-monotone paths on $P$, a…

Let $P_c=\{x\mid Bx=c,\ x\geq 0\}$ be a non-degenerate $m\times n$ classical transportation polytope, and let its graph have the vertices and edges of $P_c$. Hamiltonian-cycle conj…