The degree and codegree threshold for linear triangle covering in 3-graphs

Abstract

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 ${\delta }_i(G)>c_i(n,F)$ 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 $\{v_1{v}_2{v}_3,v_3{v}_4{v}_5,v_5{v}_6{v}_1\}$.

10 pages, 1 figure