# Concatenating bipartite graphs

## TL;DR

This paper studies a function related to bipartite graph concatenation, exploring its properties, symmetries, and discontinuities, and raises open questions about its behavior.

## Contribution

It introduces the function (x,y) for bipartite graph concatenation, analyzes its properties, symmetry, and points of discontinuity, and proposes new questions and conjectures.

## Key findings

- (x,y) is symmetric in x and y
- (x,y) has a discontinuity at x=y=1/k for integers k>1
- The paper raises open questions about the function's properties

## Abstract

Let $x,y\in(0,1]$ and let $A,B,C$ be disjoint nonempty 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 $C$. 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. The function $\phi$ is interesting, and we investigate some of its properties. For instance, we show that it is symmetric in $x$ and $y$, and that it has a discontinuity at $x=y=1/k$ for all integers $k>1$. We raise a number of questions and conjectures.