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| Two Extremal Problems in LinearHypergraphs |
| Guorong Gao,Yi Long,Lusheng Fang |
| (Center for Discrete Mathematics and Theoretical Computer Science,
Fuzhou University, Fuzhou 350108, China;
School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108,
China) |
| DOI： |
| Abstract: |
| An r-uniform hypergraph is linear if any pair of edges of the hypergraph has at most one common vertex. In this paper, we focus on Tur′an
type problems for linear hypergraphs. For a graph F, the r-expansion F
+ is the
r-graph obtained from F by enlarging each edge of F with r?2 new vertices
disjoint from V (F) such that distinct edges of F are enlarged by distinct vertices. First, we prove that a Ks,t
+ -free r-partite linear r-graph of order n has at
most t?1
r?1
1
s
n
2? 1
s +O(n
2? 2
s ) edges, which strengthens the results of Lazebnik,
Verstra¨ete [Electron. J. Comb., 10(2003), #R25] and Timmons [Electron. J.
Combin. 29 (3)(2022)#P3.46]. Second, we give a sharp upper bound for the
number of edges of linear hypergraphs containing no expansion of double star,
where the double star is a tree with two vertices of degree greater than one. |
| Key words: Linear hypergraph, Tur´an problem, Bipartite graph, r-partite |