Triangle-degrees in graphs and tetrahedron coverings in 3-graphs

DOI10.1017/S0963548320000061zbMATH Open1466.05180arXiv1901.09560MaRDI QIDQ4993257

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Publication date: 15 June 2021

Published in: (Search for Journal in Brave)

Abstract: We investigate a covering problem in $3$-uniform hypergraphs ($3$-graphs): given a $3$-graph $F$, what is $c_1(n,F)$, the least integer $d$ such that if $G$ is an $n$-vertex $3$-graph with minimum vertex degree $delt{a}_1(G)>d$ then every vertex of $G$ is contained in a copy of $F$ in $G$ ? We asymptotically determine $c_1(n,F)$ when $F$ is the generalised triangle $K_4^{(3)-}$, and we give close to optimal bounds in the case where $F$ is the tetrahedron $K_4^{(3)}$ (the complete $3$-graph on $4$ vertices). This latter problem turns out to be a special instance of the following problem for graphs: given an $n$-vertex graph $G$ with $m>n^2/4$ edges, what is the largest $t$ such that some vertex in $G$ must be contained in $t$ triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.

Full work available at URL: https://arxiv.org/abs/1901.09560

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