关于图的亏数问题
On deficiency problems for graphs
摘要：
受Steiner三元系和拉丁方的类似问题的启发，Nenadov，Sudakov和Wagner [Completion and deficiency problems，Journal of Combinatorial Theory Series B，2020]最近引入了图亏的概念。给定全局生成性质$\mathcal P$和图$G$，图$G$关于性质$\mathcal P$的亏数$\text{def}(G)$是使得联接$G*K_t$具有性质$\mathcal P$的最小非负整数$t$。特别地，Nenadov，Sudakov和Wagner提出了一个问题，即确定$n$ -顶点图$G$需要多少条边来确保$G*K_t$包含$K_r$ -因子（对于任何固定的$r\geq 3$）.在本文中，我们完全解决他们的问题。我们也给出了一个类似的结果，迫使$G*K_t$包含任何固定的二部$(n+t)$ -顶点图的有界度和小带宽。
英文摘要：
Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $\mathcal P$ and a graph $G$, the deficiency $\text{def}(G)$ of the graph $G$ with respect to the property $\mathcal P$ is the smallest non-negative integer $t$ such that the join $G*K_t$ has property $\mathcal P$. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an $n$-vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$-factor (for any fixed $r\geq 3$). In this paper we resolve their problem fully. We also give an analogous result which forces $G*K_t$ to contain any fixed bipartite $(n+t)$-vertex graph of bounded degree and small bandwidth.
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。
