The list coloring number of uncrowded hypergraphs

Summary · qwen2.5:32b

The study proves that for any fixed integer \(r \geq 2\) and any \(\varepsilon > 0\), sufficiently large finite uncrowded \((r+1)\)-uniform hypergraphs with maximum degree \(\Delta\) have a list chromatic number at most \((1+\varepsilon)\left(\frac{r\Delta}{\log\Delta}\right)^{1/r}\). This result is significant as it provides an upper bound for the list coloring number of such hypergraphs, utilizing a semi-random nibble technique followed by a Rosenfeld-style counting argument to complete the coloring.

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Learn about a new proof for list coloring number of uncrowded hypergraphs in the field of rendering and mathematics.

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arXiv:2607.05256v1 Announce Type: new Abstract: We prove that for every fixed integer $r\geq 2$ and every $\varepsilon>0$, every sufficiently large finite uncrowded $(r+1)$-uniform hypergraph of maximum degree $\Delta$ has list chromatic number at most \[ (1+\varepsilon)\left(\frac{r\Delta}{\log\Delta}\right)^{1/r}. \] The proof is a semi-random list-coloring nibble carried out directly on the original hypergraph. We encode the remaining coloring problem by active edge-color constraints and control all residual sizes through a binomial degree bound. After the nibble reaches a sparse terminal state, the coloring is completed by a Rosenfeld-style counting argument.