Degrees in link graphs of regular graphs

TL;DR

This paper investigates extremal properties of link graphs in regular graphs, establishing tight bounds on their minimum degrees and exploring related problems involving subgraphs induced by spheres and balls.

Contribution

It provides the first tight bounds on the minimum degree of link graphs in regular graphs and extends the analysis to subgraphs induced by balls, motivated by expansion conjectures.

Findings

Some link graph has minimum degree at most  d/3-1 for connected non-complete regular graphs.

For large graphs, some link graph has minimum degree at most  d/2-1.

Results are tight bounds, with implications for expansion properties in bounded-degree graphs.

Abstract

We analyse an extremal question on the degrees of the link graphs of a finite regular graph, that is, the subgraphs induced by non-trivial spheres. We show that if $G$ is $d$-regular and connected but not complete then some link graph of $G$ has minimum degree at most $\lfloor 2d/3\rfloor-1$, and if $G$ is sufficiently large in terms of $d$ then some link graph has minimum degree at most $\lfloor d/2\rfloor-1$; both bounds are best possible. We also give the corresponding best-possible result for the corresponding problem where subgraphs induced by balls, rather than spheres, are considered. We motivate these questions by posing a conjecture concerning expansion of link graphs in large bounded-degree graphs, together with a heuristic justification thereof.