On $2$-factors of Hamiltonian graphs

TL;DR

This paper proves that large Hamiltonian graphs with very small minimum degree contain specific 2-factors with exactly k cycles, advancing understanding of graph cycle structures under minimal degree conditions.

Contribution

It establishes the first polynomially smaller than linear minimum-degree condition ensuring the existence of a 2-factor with exactly k cycles in large Hamiltonian graphs.

Findings

Large Hamiltonian graphs with minimum degree n^{1-ε} contain a 2-factor with exactly k cycles.

The result applies even when the graph only contains a 2-factor with at most k cycles.

The methods answer a question about 2-factors in graphs with relaxed Hamiltonian conditions.

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 Buci\'c, Jahn, Pokrovskiy and Sudakov.