关于无$K_{2,t}$图的编辑距离（扩充版）
On the edit distance from $K_{2,t}$-free graphs (Extended Version)
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边缘密度$p$与$\mathcal{H}$的最远图。本文研究了一类两点完全二部图的遗传性质$\forb(K_{2,t})$，即一类两点完全二部图没有导出副本，另一类两点完全二部图有$t$.采用各种各样的技术和彩色正则图的结构，我们能够确定编辑距离函数在整个域$p\in [0,1]$时$t=3,4$和扩展的间隔，其中编辑距离函数为$\forb(K_{2,t})$是已知的所有值的$t$，确定其最大值为所有奇数$t$。我们还证明了奇$t$的功能有一个非平凡的区间上，它达到了最大值。这些是唯一已知的主要遗传特性，这种情况发生。在研究这类函数的过程中，我们遇到了一些令人惊讶的联系，极值图论问题，如强正则图和问题的Zarankiewicz。这是一篇同名论文的扩展版本，现在发表在《图论杂志》\cite{jgt_version}上。特别是，这个版本包含附录A，其中有关于小$t$的遗传性质$\forb(K_{2,t})$的表格和图表，以及附录B，其中有引理31，命题32，命题33和引理34的证明。
英文摘要：
The edit distance between two graphs on the same vertex set is defined to be the 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 of edge density $p$ from $\mathcal{H}$ under this metric. In this paper, we address the hereditary property $\forb(K_{2,t})$, the property of having no induced copy of the complete bipartite graph with 2 vertices in one class and $t$ in the other. Employing an assortment of techniques and colored regularity graph constructions, we are able to determine the edit distance function over the entire domain $p\in [0,1]$ when $t=3,4$ and extend the interval over which the edit distance function for $\forb(K_{2,t})$ is known for all values of $t$, determining its maximum value for all odd $t$. We also prove that the function for odd $t$ has a nontrivial interval on which it achieves its maximum. These are the only known principal hereditary properties for which this occurs. In the process of studying this class of functions, we encounter some surprising connections to extremal graph theory problems, such as strongly regular graphs and the problem of Zarankiewicz. This is an extended version of a paper with the same name now published in the Journal of Graph Theory \cite{jgt_version}. In particular, this version contains Appendix A, which has tables and graphs pertaining to the hereditary property $\forb(K_{2,t})$ for small $t$, and Appendix B, which has the proofs of Lemma 31, Proposition 32, Proposition 33, and Lemma 34.
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