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Area of Science:

  • Graph theory
  • Probability theory
  • Random graph models

Background:

  • Introduces 1-independent probability measures (1-ipms) for graph edge sets, where independent events are at graph distance at least 1.
  • Defines random graph models associated with 1-ipms and the collection of 1-ipms where edge inclusion probability is at least p.
  • Highlights the problem posed by Balister and Bollobás regarding the critical probability p* for the emergence of infinite components in random graphs.

Purpose of the Study:

  • To improve lower bounds on the critical probability p* for the emergence of infinite components in random graphs under 1-ipms.
  • To determine the 1-independent critical probability for the formation of long paths in line and ladder lattices.
  • To study the infimum probability of graph connectivity (f_1,(p)) for finite graphs under 1-ipms.

Main Methods:

  • Utilizes advanced techniques in probability theory and graph theory to analyze random graph models.
  • Develops new methods to establish improved lower bounds for critical probabilities.
  • Applies specific calculations for path, complete, and cycle graphs to determine connectivity probabilities.

Main Results:

  • Significantly improves existing lower bounds for p*, the critical probability for infinite components.
  • Determines the exact 1-independent critical probability for the emergence of long paths on line and ladder lattices.
  • Provides exact values for f_1,(p), the infimum connectivity probability, for path, complete, and cycle graphs (up to length 5).

Conclusions:

  • The research advances the understanding of random graph properties governed by 1-independent probability measures.
  • New bounds and exact results offer valuable insights into percolation phenomena and connectivity in specific graph structures.
  • The findings contribute to the theoretical framework of random graphs and their critical behaviors.