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arXiv · 2605.11288

On $2$-factors of Hamiltonian graphs

Alberto Espuny Díaz·António Girão·Bertille Granet·Gal Kronenberg
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Abstract

Let $k\geq 2$. We show that, for a sufficiently small $\varepsilon>0$, any sufficiently large $n$-vertex Hamiltonian graph of minimum degree at least $n^{1-\varepsilon}$ contains a $2$-factor consisting of exactly $k$ cycles. This is the first minimum-degree condition which is polynomially smaller than linear. Our methods yield an analogous result when the host graph is not required to contain a Hamilton cycle, but only a $2$-factor consisting of at most $k$ cycles; this answers a question of Bucić, Jahn, Pokrovskiy and Sudakov.

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Alberto Espuny Díaz, António Girão, Bertille Granet, Gal Kronenberg. 2026-05-11. On $2$-factors of Hamiltonian graphs. https://arxiv.org/abs/2605.11288

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