Some results on concatenating bipartite graphs

TL;DR

This paper extends previous analysis of functions related to bipartite graph concatenations, determining conditions under which certain connectivity thresholds are achieved in complex bipartite structures.

Contribution

It advances the understanding of bipartite graph concatenation functions by establishing new bounds for when these functions exceed specific thresholds.

Findings

Extended bounds for $ heta(x,y)$ and $ ho(x,y)$ functions.

Identified new parameter ranges where the functions surpass 3/4, 2/5, and 3/5.

Built upon prior results to deepen the theoretical understanding of bipartite graph concatenations.

Abstract

We consider two functions $\phi$ and $\psi$, defined as follows. 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 $x|B|$ neighbors in $B$, and every vertex in $B$ has at least $y|C|$ neighbors 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. If in addition we require that every vertex in $B$ has at least $x|A|$ neighbors in $A$, and every vertex in $C$ has at least $y|B|$ neighbors in $C$, we denote by $\psi(x,y)$ 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. In their recent paper, M. Chudnovsky, P. Hompe, A. Scott, P. Seymour, and S. Spirkl introduced these functions, proved some general results about them, and analyzed when they are greater than or equal to $1/2, 2/3,$ and $1/3$. Here, we extend their results by analyzing when they are greater than or equal to $3/4, 2/5,$ and $3/5$.