We consider the symmetric difference of two graphs on the same vertex set $[n]$, which is the graph on $[n]$ whose edge set consists of all edges that belong to exactly one of the two graphs. Let $\mathcal{F}$ be a class of graphs, and let $M_{\mathcal{F}}(n)$ denote the maximum possible cardinality of a family $\mathcal{G}$ of graphs on $[n]$ such that the symmetric difference of any two members in $\mathcal{G}$ belongs to $\mathcal{F}$. These concepts are recently investigated by Alon, Gujgiczer, K\"{o}rner, Milojevi\'{c}, and Simonyi, with the aim of providing a new graphic approach to coding theory. In particular, $M_{\mathcal{F}}(n)$ denotes the maximum possible size of this code. Existing results show that as the graph class $\mathcal{F}$ changes, $M_{\mathcal{F}}(n)$ can vary from $n$ to $2^{(1+o(1))\binom{n}{2}}$. We study several phase transition problems related to $M_{\mathcal{F}}(n)$ in general settings and present a partial solution to a recent problem posed by Alon et. al.

翻译：我们考虑同一顶点集$[n]$上两个图的对称差，即顶点集为$[n]$、边集由恰好属于这两个图中之一的边构成的图。设$\mathcal{F}$为一类图，并令$M_{\mathcal{F}}(n)$表示定义在$[n]$上的图族$\mathcal{G}$的最大可能基数，使得$\mathcal{G}$中任意两个成员的对称差均属于$\mathcal{F}$。这些概念近期由Alon、Gujgiczer、Kőrner、Milojević和Simonyi提出，旨在为编码理论提供一种新的图论方法。特别地，$M_{\mathcal{F}}(n)$表示此码的最大可能大小。现有结果表明，随着图类$\mathcal{F}$的变化，$M_{\mathcal{F}}(n)$的取值范围可从$n$跨越至$2^{(1+o(1))\binom{n}{2}}$。我们研究了关于$M_{\mathcal{F}}(n)$在一般设置下的若干相变问题，并对Alon等人近期提出的一个难题给出了部分解答。