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arXiv · 2611.08432

An improved minimum degree threshold for fractional clique decompositions in partite graphs

Sean Longbrake·Michael Simkin·Liana Yepremyan·Michael Zheng
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Abstract

We prove a Dirac-type condition for an $r$-partite graph to admit a fractional clique decomposition: every sufficiently large $r$-partite graph with $n$ vertices in each part and partite minimum degree at least $(1 - \frac{1}{30r^2}) n$ admits a fractional $K_r$-decomposition, subject to necessary divisibility conditions. This implies the same upper bound on the asymptotic minimum degree threshold for $r$-partite graphs to admit a $K_r$-decomposition, using a well-known reduction.

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Sean Longbrake, Michael Simkin, Liana Yepremyan, Michael Zheng. 2026-11-05. An improved minimum degree threshold for fractional clique decompositions in partite graphs. https://arxiv.org/abs/2611.08432

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