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How likely is an LLD degree sequence to be graphical? (English) Zbl 1079.05023
Let \(D(1),\dots, D(n)\) be a sequnce of independent identically distributed positive integer-valued random variables, and let \(P(n)\) be the probability that the sequence is graphical, i.e. that there is a simple graph on \(n\) vertices with degrees given by the \(n\) values in the sequence. By investigating the limit of \(nP(D(i)>n-1)\) as \(n\) tends to infinity, sufficient conditions are obtained that \(P(n)\) has a limit \(0\) or \(1/2\) or strictly in between. The proof is based on a representation of order statistics by unit exponential random variables.

05C07 Vertex degrees
05C80 Random graphs (graph-theoretic aspects)
60G70 Extreme value theory; extremal stochastic processes
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