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Most edge-orderings of \(K_{n}\) have maximal altitude. (English) Zbl 1414.05268
Summary: Suppose the edges of the complete graph on \(n\) vertices are assigned a uniformly chosen random ordering. Let \(X\) denote the corresponding number of Hamiltonian paths that are increasing in this ordering. It was shown in a recent paper by M. Lavrov and P.-S. Loh [Random Struct. Algorithms 48, No. 3, 588–611 (2016; Zbl 1338.05150)] that this quantity is nonzero with probability at least \(1/e - o(1)\), and conjectured that \(X\) is asymptotically almost surely nonzero. In this paper, we prove their conjecture. We further prove a partial result regarding the limiting behavior of \(X\), suggesting that \(X/n\) is log-normal in the limit as \(n\rightarrow\infty\). A key idea of our proof is to show a certain relation between \(X\) and its size-biased distribution. This relies heavily on estimates for the third moment of \(X\).

05C80 Random graphs (graph-theoretic aspects)
05C45 Eulerian and Hamiltonian graphs
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