A proof of the Kim-Vu sandwich conjecture

Abstract: In 2004, Kim and Vu conjectured that, when $d=\omega(\log n)$, the random $d$-regular graph $G_d(n)$ can be sandwiched with high probability between two random binomial graphs $G(n,p)$ with edge probabilities asymptotically equal to $\frac{d}{n}$. That is, there should exist $p_=(1-o(1))\frac{d}{n}$, $p^=(1+o(1))\frac{d}{n}$ and a coupling $(G_,G,G^)$ such that $G_\sim G(n,p_)$, $G\sim G_d(n)$, $G^*\sim G(n,p^*)$, and $\mathbb{P}(G_\subset G\subset G^)=1-o(1)$. Known as the sandwich conjecture, such a coupling is desirable as it would allow properties of the random regular graph to be inferred from those of the more easily studied binomial random graph. The conjecture was recently shown to be true when $d\gg\log^4n$ by Gao, Isaev and McKay. In this paper, we prove the sandwich conjecture in full. We do so by analysing a natural coupling procedure introduced in earlier work by Gao, Isaev and McKay, which had only previously been done when $d\gg n/\sqrt{\log n}$.

• The paper proves the Kim-Vu Sandwich Conjecture by rigorously coupling random d-regular graphs with binomial models using comprehensive switching techniques.
• It establishes edge addition and removal processes that maintain the uniformity of d-regular graph constructions and validate typical properties with high probability.
• Numerical analyses on cycle distributions and concentration inequalities provide practical bounds, advancing both theoretical insights and real-world applications in graph theory.

A Proof of the Kim-Vu Sandwich Conjecture

Introduction

The Kim-Vu sandwich conjecture, formulated in 2004, posits a relationship between random $d$-regular graphs $G_d(n)$ and binomial random graphs $G(n,p)$ with similar edge probabilities, specifically concerning their containment properties. The conjecture suggests that for sufficiently large $d$, a random $d$-regular graph can be "sandwiched" between two binomial graphs with probabilities adjusted around $d/n$. This has significant implications for inferring properties of $G_d(n)$ from $G(n,p)$, which is mathematically easier to analyze due to its independence of edges. Recent progress, particularly the work by Gao, Isaev, and McKay, culminates in proving the conjecture when $d \gg \log^4 n$. The current paper extends this by proving the conjecture in full, using comprehensive switching techniques to analyze constraints between $G_d(n)$, $G(n,p_\ast)$, and $G(n,p^\ast)$.

Methodology

To demonstrate the sandwiching effect, an edge addition and removal process is employed, carefully designed to preserve the random uniformity of the target $d$-regular graph distribution while iteratively constructing the required subgraph coupling. Critical to the success of this approach is an accurate estimation of configurations that allow potential edges to contribute to regular graphs, which is expounded through a detailed study on the distribution of cycle lengths within these graphs.

Edge Addition Process

The edge addition process begins with an empty graph and iteratively considers adding edges, conditioned on maintaining a $d$-regular graph potential. By evaluating potential $d$-regular supergraphs of the current graph state, edges are selectively added with probabilities proportional to their potential to form regular graphs:

for each edge e in possible_edges: if F+e is part of a d-regular supergraph: add e with probability proportional to regular supergraph count

Edge Removal Process

Conversely, a complementary edge removal process commences with the complete graph and selectively deletes edges to approximate $G_d(n)$, maintaining the uniformity of possible $d$-regular subgraphs:

for each edge e in current_graph_edges: if current_graph - e retains potential regular subgraphs: remove e with proportional likelihood

Proving the Sandwich Conjecture

The proof leverages switching techniques, initially analyzing small cycles and paths within intended regular subgraphs. By employing a bipartite switching framework, transitions between configurations (states) are calculated, bounding deviations in degree sequences. Concentration inequalities on these transitions are derived, ensuring that typical graph properties hold with high probability. Multivariate polynomial concentration bounds are strategically applied to ensure robustness of the methods, providing a practical pathway to realizing the conjecture's conditions.

Numerical Significance

Throughout the graph formation processes, strong numerical bounds are placed on the paths and cycles that are active, ensuring that deviation from expected regular structure remains unusually low. Key insights derived from these numerical analyses are employed to calibrate the likelihood thresholds necessary for successful coupling between the graph models.

Conclusion

The comprehensive proof furnished by this paper extends the scope of coupling random regular graphs with binomial models beyond prior boundaries, establishing a resilient framework with implications for both theoretical study and practical applications in graph theory. This resolution of the Kim-Vu sandwich conjecture will invariably influence ongoing research initiatives aiming to characterize connectivity and distribution properties of graph classes more broadly.