Large generalized books are p-good

非$r$-部图中的谱Turán型问题：禁用广义书图$B_{r,k}$

Spectral Turán-type problem in non-$r$-partite graphs: Forbidden generalized book graph $B_{r,k}$

本文针对非$r$-部图中的谱Turán型问题，利用谱稳定性理论、局部结构刻画及特征方程与Rayleigh商方程方法，对$r\geq3$、$k\geq1$和充分大的$n$确定了禁用广义书图$B_{r,k}$时唯一极图，部分解决了文献[38]中的公开问题。

书籍、轮子及一般化的小拉姆齐数

Small Ramsey numbers for books, wheels, and generalizations

本文通过多种方法确定了书籍和轮子的拉姆齐数，并研究了广义拉姆齐数的某些新界限。

非$r$-部图中广义书图$B_{r,k}$的精确Turán数

The exact Turán number of generalized book graph $B_{r,k}$ in non-$r$-partite graphs

本文基于稳定性理论和局部结构刻画，确定了在阶数$n$足够大时，非$r$-部图中广义书图$B_{r,k}$的精确Turán数及所有对应的极值图。

Some exact results for generalized Turán problems

Fix a $k$-chromatic graph $F$. In this paper we consider the question to determine for which graphs $H$ does the Turán graph $T_{k-1}(n)$ have the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large enough). We say that such a graph $H$ is $F$-Turán-good. In addition to some general results, we give (among others) the following concrete results: (i) For every complete multipartite graph $H$, there is $k$ large enough such that $H$ is $K_k$-Turán-good. (ii) The path $P_3$ is $F$-Turán-good for $F$ with $χ(F) \geq 4$. (iii) The path $P_4$ and cycle $C_4$ are $C_5$…

Paths of Length Three are $K_{r+1}$-Turán Good

The generalized Turán problem $ext(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Turán problem is often the original Turán graph. They gave the name "$F$-Turán-good" to graphs $T$ for which, for large enough $n$, the solution to the generalized Turán problem is realized by a Turán graph. They prove that the path graph on two edges, $P_2$, is $K_{r+1}$-Turán-good for all $r \ge 3$, but they conjecture that the same result should hold for all $P_\ell$…

Some exact results of the generalized Turán numbers for paths

For graphs $H$ and $F$ with chromatic number $χ(F)=k$, we call $H$ strictly $F$-Turán-good (or $(H, F)$ strictly Turán-good) if the Turán graph $T_{k-1}(n)$ is the unique $F$-free graph on $n$ vertices containing the largest number of copies of $H$ when $n$ is large enough. Let $F$ be a graph with chromatic number $χ(F)\geq 3$ and a color-critical edge and let $P_\ell$ be a path with $\ell$ vertices. Gerbner and Palmer (2020, arXiv:2006.03756) showed that $(P_3, F)$ is strictly Turán good if $χ(H)\ge 4$ and they conjectured that (a) this result is true when $χ(F)=3$, and, moreover, (b) $(P_\el…

On the Nonexistence of Some Generalized Folkman Numbers

For an undirected simple graph $G$, we write $G \rightarrow (H_1, H_2)^v$ if and only if for every red-blue coloring of its vertices there exists a red $H_1$ or a blue $H_2$. The generalized vertex Folkman number $F_v(H_1, H_2; H)$ is defined as the smallest integer $n$ for which there exists an $H$-free graph $G$ of order $n$ such that $G \rightarrow (H_1, H_2)^v$. The generalized edge Folkman numbers $F_e(H_1, H_2; H)$ are defined similarly, when colorings of the edges are considered. We show that $F_e(K_{k+1},K_{k+1};K_{k+2}-e)$ and $F_v(K_k,K_k;K_{k+1}-e)$ are well defined for $k \geq 3$. …