Critical threshold for regular graphs

2024/12/01 by Ishaan Bhadoo, Bhadoo, Ishaan
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2412.00635

openalex publication_date 2024/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we study the critical percolation threshold p_c for d-regular graphs. It is well-known that p_c ≥ (1)/(d-1) for such graphs, with equality holding for the d-regular tree. We prove that among all quasi-transitive d-regular graphs, the equality p_c(G) = (1)/(d-1) holds if and only if G is a tree. Furthermore, we provide counterexamples that illustrate the necessity of the quasi-transitive assumption.

Related

• Is the critical percolation probability local?
• Critical percolation on random regular graphs
• Strict monotonicity of percolation thresholds under covering maps
• Sharp threshold for K₄-percolation
• Components, large and small, are as they should be II: supercritical percolation on regular graphs of constant degree