Long paths and connectivity in 1‐independent random graphs

AN Day, V Falgas‐Ravry… - Random Structures & …, 2020 - Wiley Online Library
Random Structures & Algorithms, 2020Wiley Online Library
A probability measure on the subsets of the edge set of a graph G is a 1‐independent
probability measure (1‐ipm) on G if events determined by edge sets that are at graph
distance at least 1 apart in G are independent. Given a 1‐ipm, denote by the associated
random graph model. Let denote the collection of 1‐ipms on G for which each edge is
included in with probability at least p. For, Balister and Bollobás asked for the value of the
least p⋆ such that for all p> p⋆ and all, almost surely contains an infinite component. In this …
A probability measure on the subsets of the edge set of a graph G is a 1‐independent probability measure (1‐ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1‐ipm , denote by the associated random graph model. Let denote the collection of 1‐ipms on G for which each edge is included in with probability at least p. For , Balister and Bollobás asked for the value of the least p such that for all p > p and all , almost surely contains an infinite component. In this paper, we significantly improve previous lower bounds on p. We also determine the 1‐independent critical probability for the emergence of long paths on the line and ladder lattices. Finally, for finite graphs G we study f1, G(p), the infimum over all of the probability that is connected. We determine f1, G(p) exactly when G is a path, a complete graph and a cycle of length at most 5.
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