阈值图及更广泛类别的拉姆齐型结果
Ramsey-type results for threshold graphs and beyond
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中文总结 AI 辅助
该研究针对阈值图定义了$r'_2(s)$等拉姆齐型参数，确定其指数界与部分精确值，还推导了完全二部图对应参数的渐近公式及部分精确值。
中文摘要 AI 辅助
阈值图是一种可通过从单顶点图出发，反复添加支配顶点或孤立顶点构造而成的图。受该类图的诱导拉姆齐型问题驱动，我们定义$r'_2(s)$为最小整数$n$，使得每个$n$顶点图都包含一个$s$顶点的诱导阈值图。我们确定了$r'_2(s)$的指数上下界，并给出$s \in\{3,4,5,6\}$时的精确值。为从边着色视角研究该问题，我们使用Richer在《J. Combin. Theory Ser. B》80(1) (2000), 172--177中引入的可序着色概念：边着色图是可序的，当且仅当可对其顶点排序，使得每个顶点到后续顶点的所有边颜色相同。等价地，$r'_2(s)$是最小的$n$，使得$K_n$的每个2边着色都包含一个可序$K_s$。我们还确定了所有$s \ge 3$时无序典范拉姆齐数$CR(s,3)$的精确值，其中$CR(s,3)$指最小整数$n$，使得$K_n$的每个边着色都包含一个可序$K_s$或一个彩虹$K_3$。更一般地，对图$G$和$H$，我们研究$r'_2(G)$（对应可序$G$的2色拉姆齐数）和$CR(G,H)$（备选为彩虹$H$）。对于完全二部图，我们证明对每个固定$s$，当$t \to\infty$时，$r'_2(K_{s,t}) = CR(K_{s,t}, K_3)= \left(\frac{2^s}{s+1}+o(1)\right)t$。对$s \in\{2,3\}$，我们利用强正则图、阿达马矩阵和会议矩阵构造，进一步确定了无穷多$t$值下这些参数的精确值。
英文摘要
A {\it threshold graph} is a graph that can be constructed from the one-vertex graph by repeatedly adding either a dominating vertex or an isolated vertex. Motivated by an induced Ramsey-type problem for this class, we define $r'_2(s)$ to be the minimum integer $n$ such that every $n$-vertex graph contains an induced threshold graph on $s$ vertices. We establish exponential upper and lower bounds for $r'_2(s)$ and determine its exact values for $s\in\{3,4,5,6\}$. To study this problem from an edge-coloring perspective, we use the notion of an orderable coloring, introduced by Richer [{\it J. Combin. Theory Ser. B}, 80(1) (2000), 172--177]. An edge-colored graph is {\it orderable} if its vertices can be ordered so that, for each vertex, all edges from it to later vertices have the same color. Equivalently, $r'_2(s)$ is the minimum $n$ such that every $2$-edge-coloring of $K_n$ contains an orderable $K_s$. We also determine the exact value of the unordered canonical Ramsey number $CR(s, 3)$ for all $s \ge 3$, where $CR(s,3)$ denotes the minimum integer $n$ such that every edge-coloring of $K_n$ contains either an orderable $K_s$ or a rainbow $K_3$. More generally, for graphs $G$ and $H$, we study $r'_2(G)$, the corresponding $2$-color Ramsey number for an orderable $G$, and $CR(G,H)$, where the alternative is a rainbow $H$. For complete bipartite graphs, we prove that for every fixed $s$, $r'_2(K_{s,t}) = CR(K_{s,t}, K_3)= \left(\frac{2^s}{s+1}+o(1)\right)t$ as $t\to\infty$. For $s\in \{2,3\}$, we further determine the exact values of these parameters for infinitely many $t$, using constructions arising from strongly regular graphs, Hadamard matrices and conference matrices.
