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arXiv · 2607.05027

Edge-disjoint Hamilton cycles under a bipartite-hole condition

Yanan Hu·Chengli Li·Feng Liu
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Abstract

In 2017, McDiarmid and Yolov introduced the bipartite-hole-number $\widetilde{\alpha}(G)$ and proved that $\delta(G)\ge \widetilde{\alpha}(G)$ forces a Hamilton cycle. They also gave a sufficient condition for packing edge-disjoint Hamilton cycles, and asked whether this condition is sharp or can be relaxed. For integers $a,k\ge 2$, let $f(a,k)$ be the least integer $d$ such that every graph $G$ on at least three vertices with $\widetilde{\alpha}(G)\le a$ and $\delta(G)\ge d$ contains $k$ pairwise edge-disjoint Hamilton cycles. We prove that $f(a,k)=\Theta\left(a+k+\frac{ak}{\log(k+2)}\right).$ The upper bound uses a deletion lemma for the bipartite-hole-number together with the McDiarmid--Yolov Hamiltonicity theorem and a greedy packing argument. The lower bound is obtained from three extremal constructions, the logarithmic one using a sparse random auxiliary graph with no prescribed bipartite hole.

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Yanan Hu, Chengli Li, Feng Liu. 2026-07-06. Edge-disjoint Hamilton cycles under a bipartite-hole condition. https://arxiv.org/abs/2607.05027

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