Analogues of Bermond-Bollobás Conjecture for Cages Yield Expander Families

Abstract

This paper presents a possible link between Cages and Expander Graphs by introducing three interconnected variants of the Bermond and Bollobás Conjecture, originally formulated in 1981 within the context of the Degree/Diameter Problem. We adapt these conjectures to cages, with the most robust variant posed as follows: Does there exist a constant $c$ such that for every pair of parameters $(k,g)$ there exists a $k$-regular graph of girth $g$ and order not exceeding $M(k,g)+c$?; where $M(k,g)$ denotes the value of the so-called Moore bound for cages. We show that a positive answer to any of the three variants of the Bermond and Bollobás Conjecture for cages considered in our paper would yield expander graphs (expander families); thereby establishing a connection between Cages and Expander Graphs.