发表机构
Colorado State University（科罗拉多州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对局部稀疏图，推广了局部Shearer界并改进了独立多项式的相关界，拓展了独立集与分数着色的随机构造方法。
AI 中文摘要
我们分析局部稀疏图中独立集的随机构造，这类图具体指邻域具有有界最大平均度或邻域为分数r-可着色的图。将方法专门用于寻找大独立集和低权分数着色时，我们聚焦于优化边际，但也通过优化熵推导得到能找到大量独立集的结果（即给出独立多项式的下界）。我们的主要结果将Martinsson与Steiner针对无三角形图的局部Shearer界，推广到含少量三角形的图及邻域为分数r-可着色的图，在后一种情形中改进了Dhawan的结果。我们还将通过归纳得到的独立多项式界推广到这类图，通过将必要假设从最大度条件放宽为平均度条件，改进了通过局部占用得到的已知界。
英文摘要
We analyze random constructions of independent sets in locally sparse graphs, specifically graphs with bounded maximum average degree in neighborhoods or with fractionally $r$-colorable neighborhoods. Specializing our methods to finding large independent sets and low-weight fractional colorings, we focus on optimizing for marginals, but we also derive results that find many independent sets (i.e.\ give lower bounds on the independence polynomial) by optimizing for entropy. Our main results generalize the local Shearer bound of Martinsson and Steiner for triangle-free graphs to graphs with few triangles and to graphs with fractionally $r$-colorable neighborhoods, in the latter case improving upon a result of Dhawan. We also extend independence polynomial bounds obtained via induction to such graphs, improving upon known bounds obtained by local occupancy by relaxing the necessary hypotheses from a maximum degree condition to an average degree condition.
Comments31 pages
