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arXiv · 2512.23886

Powers of Hamiltonian cycles in randomly augmented P\'osa-Seymour graphs

Sylwia Antoniuk·Andrzej Dudek·Andrzej Ruciński
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Abstract

We study the question of the least number of random edges that need to be added to a P\'osa-Seymour graph, that is, a graph with minimum degree exceeding $\frac k{k+1}n$, to secure the existence of the $m$-th power of a Hamiltonian cycle, $m>k$. It turns out that, depending on $k$ and $m$, this quantity may be captured by two types of thresholds, with one of them, called over-threshold, becoming dominant for large $m$. Indeed, for each $k\ge2$ and $m>m_0(k)$, we establish asymptotically tight lower and upper bounds on the over-thresholds (provided they exist) and show that for infinitely many instances of $m$ the two bounds coincide. In addition, we also determine the thresholds for some small values of $k$ and $m$.

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Sylwia Antoniuk, Andrzej Dudek, Andrzej Ruciński. 2025-12-29. Powers of Hamiltonian cycles in randomly augmented P\'osa-Seymour graphs. https://arxiv.org/abs/2512.23886

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