强（非诱导）Turán数
The strong (non-induced) Turán numbers
浏览论文内容
中文总结 AI 辅助
本文提出图不变量st(n,G)，刻画不含严格包含G的k顶点子图的最大边数，建立与经典Turán数的关系，并给出渐近估计及若干精确值。
中文摘要 AI 辅助
本文引入并探讨了以下新的图不变量：对于k个顶点上的图G（G ≠ K_k），令st(n,G)表示n阶图中不包含任何严格包含G的k顶点子图的最大边数。通过以下方式建立了与经典Turán数的基本关系：对于k个顶点上的图G，令D(G) = { H: |H| = |G|, H = G + e }。利用这一概念，我们证明了对所有n ≥ |G|，有ex(n,G) ≤ st(n,G) = ex(n, D(G)) ≤ min { ex(n,H): H ∈ D(G) }。族D(G)恰好能平滑地适用于经典极值结果，在许多情况下使我们能够获得st(n,G)的渐近精确估计以及精确值。从本文证明的众多结果中，我们列举以下作为说明。(1) 若χ(G) ≥ 3且χ(D(G)) = χ(G)，则st(n,G) = (1+o(1))ex(n,K_{χ(G)})。(2) 若χ(D(G)) = χ(G) + 1，则G是完全χ(G)-部图，且对充分大的n，st(n,G) = ex(n,K_{χ(G)+1})。(3) 对于奇数k，k ≥ 5，对充分大的n，st(n,C_k) = ex(n,C_k) = ex(n,K_3)。(4) 若T是阶为q、直径为k ≥ 2且q ≥ k+1 ≥ 3的树，则ex(n, {C_3,...,C_{k+1}}) ≤ st(n,T) ≤ ex(n, {C_3,...,C_{k+1}}) + (q-1)n。此外，我们获得了许多关于偶圈、theta图、稠密二部图以及形如G = G^* ∪ tK_1的图的结果，并且计算了所有至多4个顶点上的图的st(n,G)值。
英文摘要
In this paper we introduce and explore the following new graph invariant: For a graph $G$ on $k$ vertices, $G \neq K_k$, let $st(n,G)$ denote the maximum number of edges in a graph of order $n$ which does not contain any subgraph on $k$ vertices strictly containing $G$. A basic relation to classical Turán numbers is developed via the following: For $G$ on $k$ vertices, let $D(G) = \{ H : |H| = |G|, H = G + e \}$. Using this notion we prove that $ex(n,G) \leq st(n,G) = ex(n, D(G) ) \leq \min \{ ex(n,H) : H \in D(G) \}$ holds for all $n \geq |G|$. The family $D(G)$ happened to be smoothly amenable to the use of classical extremal results, and in many cases allows us to get asymptotically sharp estimates as well as exact values of $st(n,G)$. From the many results proved here we state the following as an illustration. (1) If $χ(G) \geq 3$ and $χ(D(G)) = χ(G)$, then $st(n,G) = (1+o(1))ex(n,K_{χ(G)})$. (2) If $χ(D(G)) = χ(G) +1$, then $G$ is a complete $χ(G)$-partite graph and $st(n,G) = ex(n,K_{χ(G) +1})$ for $n$ sufficiently large. (3) For $k$ odd, $k\geq 5$, $st(n,C_k) = ex(n,C_k) = ex(n,K_3)$ for $n$ sufficiently large. (4) If $T$ is a tree of order $q$ with diameter $k \geq 2$ and $q \geq k+1 \geq 3$, then $ex(n, \{C_3,...,C_{k+1}\}) \leq st(n,T) \leq ex(n, \{C_3,...,C_{k+1}\}) + (q-1)n$. Many results concerning even cycles, theta graphs, dense bipartite graphs and graphs of the form $G = G^* \cup tK_1$ are obtained, moreover the value of $st(n,G)$ is computed for all graphs on at most 4 vertices.
发表机构
- University of Haifa-Oranim（海法奥拉尼姆大学）
- HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究网络阿尔弗雷德·雷尼数学研究所）
- University of Pannonia（佩奇大学）
机构由 AI 辅助整理，请以论文原文为准。
