Fractional Triangle Decompositions at Partite Minimum Degree $4n/5$

Hengzhi He; Guang Cheng

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arXiv · 2609.31758

Fractional Triangle Decompositions at Partite Minimum Degree $4n/5$

Hengzhi He·Guang Cheng

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Abstract

We prove that every triangle-divisible balanced tripartite graph whose vertex classes have size $n$ and whose partite minimum degree is at least $4n/5$ admits a fractional triangle decomposition. Combined with the multipartite decomposition theorem of Barber, Kühn, Lo, Osthus, and Taylor, this implies that, for every fixed $\varepsilon<1/5$ and all sufficiently large $n$, every $\varepsilon$-dense partial Latin square of order $n$ is completable, improving the previous asymptotic bound of $2/25$. The fractional decomposition theorem is finite and exact. Its proof uses a minimum-weight perfect matching to normalize an arbitrary Farkas dual weighting, followed by an explicit direct-and-two-step routing scheme whose aggregate congestion on every off-matching edge is at most one.

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Hengzhi He, Guang Cheng. 2026-08-24. Fractional Triangle Decompositions at Partite Minimum Degree $4n/5$. https://arxiv.org/abs/2609.31758

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