The degree threshold for covering with all the connected $3$-graphs with $3$ edges

Abstract

Given two $r$-uniform hypergraphs $F$ and $H$, we say that $H$ has an $F$-covering if every vertex in $H$ is contained in a copy of $F$. Let $c_i(n,F)$ be the least integer such that every $n$-vertex $r$-graph $H$ with ${\delta }_i(H)>c_i(n,F)$ has an $F$-covering. Falgas-Ravry, Markstöm and Zhao (Combin. Probab. Comput., 2021) asymptotically determined $c_1(n,K_4^{(3)-})$, where $K_4^{(3)-}$ is obtained by deleting an edge from the complete $3$-graph on $4$ vertices. Later, Tang, Ma and Hou (arXiv, 2022) asymptotically determined $c_1(n,C_6^{(3)})$, where $C_6^{(3)}$ is the linear triangle, i.e. $C_6^{(3)}=([6],\{123,345,561\})$. In this paper, we determine $c_1(n,F_5)$ asymptotically, where $F_5$ is the generalized triangle, i.e. $F_5=([5],\{123,124,345\})$. We also determine the exact values of $c_1(n,F)$, where $F$ is any connected $3$-graphs with $3$ edges and $F\notin \{K_4^{(3)-},C_6^{(3)},F_5\}$.

arXiv comment17 pages, 10 figures