Long paths and connectivity in 1‐independent random graphs
AN Day, V Falgas‐Ravry… - Random Structures & …, 2020 - Wiley Online Library
AN Day, V Falgas‐Ravry, R Hancock
Random Structures & Algorithms, 2020•Wiley Online LibraryA 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 …
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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