Concatenating Bipartite Graphs
Let $x,y\in (0,1]$, and let $A,B,C$ be disjoint nonempty stable subsets of a graph $G$, where every vertex in $A$ has at least $x|B|$ neighbours in $B$, and every vertex in $B$ has at least $y|C|$ neighbours in $C$, and there are no edges between $A,C$. We denote by $\phi(x,y)$ the maximum $z$ such that, in all such graphs $G$, there is a vertex $v\in C$ that is joined to at least $z|A|$ vertices in $A$ by two-edge paths. This function has some interesting properties: we show, for instance, that $\phi(x,y)=\phi(y,x)$ for all $x,y$, and there is a discontinuity in $\phi(x,x)$ when $1/x$ is an integer. For $z=1/2, 2/3,1/3,3/4,2/5,3/5$, we try to find the (complicated) boundary between the set of pairs $(x,y)$ with $\phi(x,y)\ge z$ and the pairs with $\phi(x,y)<z$. We also consider what happens if in addition every vertex in $B$ has at least $x|A|$ neighbours in $A$, and every vertex in $C$ has at least $y|B|$ neighbours in $B$.
We raise several questions and conjectures; for instance, it is open whether $\phi(x,x)\ge 1/2$ for all $x>1/3$.