2023/07/04 by Ran Gu, Gu, Ran, Shuaichao Wang +1 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2307.01647
openalex publication_date 2023/07/04 · openalex created_date 2023/07/07 · openalex updated_date 2026/07/28
Given two k-uniform hypergraphs F and G, we say that G has an F-covering if for every vertex in G there is a copy of F covering it. For 1≤ i≤ k-1, the minimum i-degree δi(G) of G is the minimum integer such that every i vertices are contained in at least δi(G) edges. Let ci(n,F) be the largest minimum i-degree among all n-vertex k-uniform hypergraphs that have no F-covering. In this paper, we consider the F-covering problem in 3-uniform hypergraphs when F is the generalized triangle T, where T is a 3-uniform hypergraph with the vertex set \v1,v2,v3,v4,v5\ and the edge set \\v1v2v3\,\v1v2v4\,\v3v4v5\\. We give the exact value of c2(n,T) and asymptotically determine c1(n,T). We also consider the F-covering problem in 3-uniform hypergraphs when F are some trees, such as the linear k-path Pk and the star Sk. Especially, we provide bounds of ci(n,Pk) and ci(n,Sk) for k≥ 3, where i=1,2.