边替换下的图同构与阿米巴图族
Graphs isomorphisms under edge-replacements and the family of amoebas
AI总结：
本文系统研究阿米巴图族，通过定义可行边替换群 Fer(G) 建立局部阿米巴的代数判定条件，并构造多种实例证明全局阿米巴图族的丰富性。
摘要：
本文对一类称为阿米巴（amoebas）的图族进行了系统研究。阿米巴图最近出现在 Ramsey-Turán 理论背景下关于完全图边 2-着色的强制模式研究中，并在极值零和问题中发挥了重要作用。阿米巴图在以下运算中具有独特行为：设 G 为一个图，e∈E(G) 且 e'∈E(\overline{G})。如果图 G'=G-e+e' 与 G 同构，则称 G' 是由 G 通过执行一次可行边替换得到的。如果对于同一顶点集上 G 的任意两个副本 G_1、G_2，G_1 都能通过一系列可行边替换变换为 G_2，则称 G 为局部阿米巴（local amoeba）。另一方面，如果存在整数 t_0≥0，使得对所有 t≥t_0，G∪tK_1 都是局部阿米巴，则称 G 为全局阿米巴（global amoeba）。为了刻画 G 的可行边替换动力学，我们定义了一个群 Fer(G)，它满足：G 是局部阿米巴当且仅当 Fer(G)≅S_n，其中 n 是 G 的阶数。通过这一代数框架，可以更深入地理解阿米巴图的结构及其内在性质。此外，我们给出了不同的构造，证明了这些图族的丰富性，其中表明：任何连通图都可以成为某个全局阿米巴图的连通分支；全局阿米巴图可以非常稠密；并且相对于其阶数，它们可以具有很大的团数和色数。同时，还构造了一族具有类斐波那契结构且最大度任意大的全局阿米巴树。
英文摘要：
This paper offers a systematic study of a family of graphs called amoebas. Amoebas recently emerged from the study of forced patterns in $2$-colorings of the edges of the complete graph in the context of Ramsey-Turan theory and played an important role in extremal zero-sum problems. Amoebas are graphs with a unique behavior with regards to the following operation: Let $G$ be a graph and let $e\in E(G)$ and $e'\in E(\overline{G})$. If the graph $G'=G-e+e'$ is isomorphic to $G$, we say $G'$ is obtained from $G$ by performing a \emph{feasible edge-replacement}. We call $G$ a \emph{local amoeba} if, for any two copies $G_1$, $G_2$ of $G$ on the same vertex set, $G_1$ can be transformed into $G_2$ by a chain of feasible edge-replacements. On the other hand, $G$ is called \emph{global amoeba} if there is an integer $t_0 \ge 0$ such that $G \cup tK_1$ is a local amoeba for all $t \ge t_0$. To model the dynamics of the feasible edge-replacements of $G$, we define a group ${\rm Fer}(G)$ that satisfies that $G$ is a local amoeba if and only if ${\rm Fer}(G) \cong S_n$, where $n$ is the order of $G$. Via this algebraic setting, a deeper understanding of the structure of amoebas and their intrinsic properties comes into light. Moreover, we present different constructions that prove the richness of these graph families showing, among other things, that any connected graph can be a connected component of a global amoeba, that global amoebas can be very dense and that they can have, in proportion to their order, large clique and chromatic numbers. Also, a family of global amoeba trees with a Fibonacci-like structure and with arbitrary large maximum degree is constructed.
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