发表机构
Jagiellonian University（雅盖隆大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探讨随机非均匀超图的2-可着色性，发现非均匀性对存在性与算法界限影响不同，并扩展了均匀超图着色算法，构造出高权重下仍可成功着色的实例。
AI 中文摘要
我们证明，在具有几种允许边大小的随机超图中，非均匀性以根本不同的方式影响2-可着色性（Property B）的存在性和算法界限。为每条k-边赋予权重$2^{-k}$，我们根据总边权重获得了上下界，随着最小允许边大小的增长，这些界限在渐近意义上与均匀情形下的界限相匹配，而无论边如何在大小之间分布。我们还将随机均匀超图2-着色的最知名算法扩展到非均匀设置。当每条k-边赋予权重$\frac{k}{2^k}$时，我们构造了非均匀实例，其每顶点的期望总边权重可以任意大，但该算法仍能以渐近几乎必然的方式找到正确着色。在均匀设置中，一旦该量超过常数，算法以高概率失败。我们的构造使用充分分离的边大小，使得不同大小的边在执行过程中充分分离的阶段变得相关，且它们的影响本质上是独立的。
英文摘要
We show that, in random hypergraphs with several permitted edge sizes, non-uniformity affects existential and algorithmic bounds for 2-colorability (Property B) in fundamentally different ways. Assigning weight $2^{-k}$ to each $k$-edge, we obtain upper and lower bounds in terms of the total edge weight that asymptotically match the uniform bounds as the minimum permitted edge size grows, regardless of how edges are distributed among sizes. We also extend the best-known algorithm for 2-coloring random uniform hypergraphs to the non-uniform setting. With weight $\frac{k}{2^k}$ assigned to each $k$-edge, we construct non-uniform instances whose expected total edge weight per vertex is arbitrarily large, yet the algorithm finds a proper coloring asymptotically almost surely. In the uniform setting, the algorithm fails with high probability once this quantity exceeds a constant. Our construction uses sufficiently separated edge sizes, so that edges of different sizes become relevant at well-separated stages of the execution and their effects are essentially independent.
