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arXiv · 2402.02815

Approximate packing of independent transversals in locally sparse graphs

Debsoumya Chakraborti·Tuan Tran
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Abstract

Fix $\varepsilon >0$ and consider a multipartite graph $G$ with maximum degree at most $(1-\varepsilon)n$, parts $V_1,\ldots,V_k$ of the same size $n$, and where every vertex has at most $o(n)$ neighbors in any part $V_i$. Loh and Sudakov proved that any such $G$ has an independent transversal. They further conjectured that the vertex set of $G$ can be decomposed into pairwise disjoint independent transversals. In the present paper, we resolve this conjecture approximately by showing that $G$ contains $(1-\varepsilon)n$ pairwise disjoint independent transversals. As applications, we give approximate answers to questions of Yuster, and of Fischer, Kühn, and Osthus.

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Debsoumya Chakraborti, Tuan Tran. 2025-07-27. Approximate packing of independent transversals in locally sparse graphs. https://arxiv.org/abs/2402.02815

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