变形虫的递归结构
Recursive constructions of amoebas
摘要：
全局变形虫是一个广泛而丰富的图族，它是从$2$中某些拉姆齐-图兰问题的研究中出现的-$2$是完全图$K_n$的边的着色，一旦保证每种颜色中有一定数量的边，就处理不可避免的模式的出现。事实上，只要满足这样的着色约束，并且如果$n$足够大，那么可以发现每个全局变形虫嵌入在$K_n$中，使得它的每种颜色都有一半的边。更令人惊讶的是，每个二分全局变形虫$G$在每个音调变化中都是不可避免的，这意味着，对于任何一对整数$r, b$，使得$r + b $是$G$的边数，存在$K_n$同构于$G$的子图，其中$r$边在第一种颜色中，$b$边在第二种颜色中。使全局变形虫工作的特征是一个接一个的边替换，使图的结构保持不变。通过一组理论的方法，这个功能的动力学可以建模。作为一个对应的全球变形虫，“生活”在一个可能大的完全图$K_n$，我们也考虑生成子图的$K_n$具有相同的功能。为了突出其丰富性和多功能性，我们在这里提出了三种不同的递归结构的变形虫，其中两个产生有趣的家庭本身，其中一个提供了广泛的可能性。
英文摘要：
Global amoebas are a wide and rich family of graphs that emerged from the study of certain Ramsey-Turán problems in $2$-colorings of the edges of the complete graph $K_n$ that deal with the appearance of unavoidable patterns once a certain amount of edges in each color is guaranteed. Indeed, it turns out that, as soon as such coloring constraints are satisfied and if $n$ is sufficiently large, then every global amoeba can be found embedded in $K_n$ such that it has half its edges in each color. Even more surprising, every bipartite global amoeba $G$ is unavoidable in every tonal-variation, meaning that, for any pair of integers $r, b$ such that $r + b $ is the number of edges of $G$, there is a subgraph of $K_n$ isomorphic to $G$ with $r$ edges in the first color and $b$ edges in the second. The feature that makes global amoebas work are one-by-one edge replacements that leave the structure of the graph invariant. By means of a group theoretical approach, the dynamics of this feature can be modeled. As a counterpart to the global amoebas that "live" inside a possibly large complete graph $K_n$, we also consider local amoebas which are spanning subgraphs of $K_n$ with the same feature. In an effort to highlight their richness and versatility, we present here three different recursive constructions of amoebas, two of them yielding interesting families per se and one of them offering a wide range of possibilities.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
