Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge beten two connected components of G is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and lsh stating that if G is bridge-addable and Gn is a uniform n-vertex graph from G, then Gn is connected with probability at least (1 + on(1))e 1/2 . The constant e 1/2 is best possible since it is reached for the class of all forests.   prove a form of uniqueness in this statement: if G is a bridge-addable class and the random graph Gn is connected with probability close to e 1/2 , then Gn is asymptotically close to a uniform n-vertex random forest in some local sense. For example, if the probability converges to e 1/2 , then Gn converges in the sense of Benjamini-Schramm to the uniform infinite random forest F1. This result is reminiscent of so-called “stability results” in extremal graph theory, with the difference that here the stable extremum is not a graph but a graph class.
['Guillaume Chapuy', 'Guillem Perarnau']
I am writing a research paper on a idea Local convergence and stability of tight bridge-addable graph classes
[{"from": "human", "value": "I am writing a research paper on a idea Local convergence and stability of tight bridge-addable graph classes"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge beten two connected components of G is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and lsh stating that if G is bridge-addable and Gn is a uniform n-vertex graph from G, then Gn is connected with probability at least (1 + on(1))e 1/2 . The constant e 1/2 is best possible since it is reached for the class of all forests.   prove a form of uniqueness in this statement: if G is a bridge-addable class and the random graph Gn is connected with probability close to e 1/2 , then Gn is asymptotically close to a uniform n-vertex random forest in some local sense. For example, if the probability converges to e 1/2 , then Gn converges in the sense of Benjamini-Schramm to the uniform infinite random forest F1. This result is reminiscent of so-called \u201cstability results\u201d in extremal graph theory, with the difference that here the stable extremum is not a graph but a graph class."}]