Sprinkling with random regular graphs

TL;DR

This paper investigates the properties of the union of two random regular graphs, conjecturing it resembles a larger regular graph, and provides proofs for certain density cases along with formulas for subgraph counts.

Contribution

It introduces a conjecture about the union of two random regular graphs and proves it in specific density regimes, also deriving formulas for regular subgraph counts.

Findings

Union of two random regular graphs behaves like a larger regular graph in certain cases

Proved the conjecture for dense and sparse graph regimes

Derived an asymptotic formula for the expected number of spanning regular subgraphs

Abstract

We conjecture that the distribution of the edge-disjoint union of two random regular graphs on the same vertex set is asymptotically equivalent to a random regular graph of the combined degree, provided it grows as the number of vertices tends to infinity. We verify this conjecture for the cases when the graphs are sufficiently dense or sparse. We also prove an asymptotic formula for the expected number of spanning regular subgraphs in a random regular graph.