关于无$K_{2,t}$图的编辑距离II：$t\geq 5$情形
On the edit distance from $K_{2,t}$-free graphs II: Cases $t\geq 5$
摘要：
同一顶点集上的两个图之间的编辑距离定义为它们的边集的对称差的大小。遗传属性的编辑距离函数$\mathcal{H}$是$p$的函数，并且在该度量下渐近地测量具有从$\mathcal{H}$到$p$的边密度$p$的最远图。编辑距离函数已被证明是难以计算的许多遗传属性。一些令人惊讶的连接到极值图论问题，如强正则图和Zarankiewicz问题，已被发现在试图计算各种编辑距离函数。本文讨论了当$t\geq5$和$t$分别为两个顶点的完全二部图的一个类和另一个类时，完全二部图没有导出副本的遗传性质$\forb(K_{2,t})$.这项工作是作者先前一篇论文的继续。采用各种各样的技术和有色正则图的建设，我们能够延长的间隔，在此期间的编辑距离函数的遗传性是众所周知的，并确定其最大值为所有奇数$t$。我们还探讨了几种结构，以改善已知的上界的功能。
英文摘要：
The edit distance between two graphs on the same vertex set is defined to be size of the symmetric difference of their edge sets. The edit distance function of a hereditary property, $\mathcal{H}$, is a function of $p$ and measures, asymptotically, the furthest graph with edge density $p$ from $\mathcal{H}$ under this metric. The edit distance function has proven to be difficult to compute for many hereditary properties. Some surprising connections to extremal graph theory problems, such as strongly regular graphs and the problem of Zarankiewicz, have been uncovered in attempts to compute various edit distance functions. In this paper, we address the hereditary property $\forb(K_{2,t})$ when $t\geq5$, the property of having no induced copy of the complete bipartite graph with 2 vertices in one class and $t$ in the other. This work continues from a prior paper by the authors. Employing an assortment of techniques and colored regularity graph constructions, we are able to extend the interval over which the edit distance function for this hereditary property is generally known and determine its maximum value for all odd $t$. We also explore several constructions to improve upon known upper bounds for the function.
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