论文标题
线性三角形覆盖3圈的学位和代码阈值
The degree and codegree threshold for linear triangle covering in 3-graphs
论文作者
论文摘要
鉴于两个$ k $ - 均匀的超图$ f $和$ g $，我们说$ g $具有$ f $ - 覆盖，如果$ g $中的每个顶点都包含在$ f $的副本中。对于$ 1 \ le i \ le k-1 $，让$ c_i（n，f）$成为最小整数，这样每$ n $ n $ vertex $ k $ k $ -sioncraph $ g $ a $δ_i（g）> c_i（g）> c_i（n，f）$都有$ f $ - 覆盖$。 Falgas-Ravry和Zhao [用于覆盖3均匀超图的Codegree阈值，Siam J. J. Invete Math。，2016]的覆盖问题已系统地研究了。去年，Falgas-Ravry，Markström和Zhao [三角形的三角形和四面体覆盖物，3颗粒，组合，概率和计算，2021年]渐近确定的$ C_1（N，F）$ C_1（N，F）$是$ F $是全身的Triangle时。在此注释中，我们给出$ C_2（n，f）$的确切值，并非渐近地确定$ c_1（n，f）$当$ f $是线性三角形$ C_6^3 $，其中$ C_6^3 $是带有3-均匀的超透明的vertex set $ \ \ { $ \ {V_1V_2V_3，V_3V_4V_5，V_5V_6V_1 \} $。
Given two $k$-uniform hypergraphs $F$ and $G$, we say that $G$ has an $F$-covering if every vertex in $G$ is contained in a copy of $F$. For $1\le i \le k-1$, let $c_i(n,F)$ be the least integer such that every $n$-vertex $k$-uniform hypergraph $G$ with $δ_i(G)> c_i(n,F)$ has an $F$-covering. The covering problem has been systematically studied by Falgas-Ravry and Zhao [Codegree thresholds for covering 3-uniform hypergraphs, SIAM J. Discrete Math., 2016]. Last year, Falgas-Ravry, Markström, and Zhao [Triangle-degrees in graphs and tetrahedron coverings in 3-graphs, Combinatorics, Probability and Computing, 2021] asymptotically determined $c_1(n, F)$ when $F$ is the generalized triangle. In this note, we give the exact value of $c_2(n, F)$ and asymptotically determine $c_1(n, F)$ when $F$ is the linear triangle $C_6^3$, where $C_6^3$ is the 3-uniform hypergraph with vertex set $\{v_1,v_2,v_3,v_4,v_5,v_6\}$ and edge set $\{v_1v_2v_3,v_3v_4v_5,v_5v_6v_1\}$.
