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Long paths and connectivity in 1-independent random graphs
A Nicholas Day1, Victor Falgas-Ravry1, Robert Hancock2
1Institutionen för Matematik och Matematisk Statistik Umeå Universitet Umeå Sweden.
Random Structures & Algorithms
|December 17, 2020
Summary
This study investigates 1-independent probability measures (1-ipms) in random graphs. Researchers improved lower bounds for critical probabilities ensuring infinite components and determined probabilities for long paths and graph connectivity.
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.
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