The shifting method and generalized Turán number of matchings

  title={The shifting method and generalized Tur{\'a}n number of matchings},
  author={Jian Wang},
  journal={Eur. J. Comb.},
  • Jian Wang
  • Published 5 December 2018
  • Mathematics
  • Eur. J. Comb.
The Maximum Number of Cliques in Hypergraphs without Large Matchings
It is shown that $\mathcal{F}^{(r)_{n,k,a}$ maximizes the number of $s$-cliques in hypergraphs on $n$ vertices with matching number at most $k$ for sufficiently large, where a=\lfloor \frac{s-r}{k} \rfloor+1$.
The Generalized Turán Number of Spanning Linear Forests
Let $${\mathcal{F}}$$ F be a family of graphs. A graph G is called $${\mathcal{F}}$$ F -free if for any $$F\in {\mathcal{F}}$$ F ∈ F , there is no subgraph of G isomorphic to F . Given a graph T and
The Maximum Number of Copies of $K_{r, s}$ in Graphs Without Long Cycles or Paths
The circumference of a graph is the length of a longest cycle of it. We determine the maximum number of copies of $K_{r,s}$, the complete bipartite graph with classes sizes $r$ and $s$, in
Many H-Copies in Graphs with a Forbidden Tree
It is shown that if $F$ is a tree then $\operatorname{ex}(n, H, F) = \Theta(n^r)$ for some integer $r = r(H,F)$, thus answering one of their questions.
The maximum number of stars in a graph without linear forest
For two graphs J and H, the generalized Turán number, denoted by ex(n, J,H), is the maximum number of copies of J in an H-free graph of order n. A linear forest F is the disjoint union of paths. In
The maximum number of $K_{r_1,\ldots,r_s}$ in graphs with a given circumference or matching number
Let Kr1,...,rs denote the complete multipartite graph with class sizes r1, . . . , rs and let Ks denote the complete graph of order s. In 2018, Luo determined the maximum number ofKs in 2-connected
Generalized Turán Problems for Small Graphs
Abstract For graphs H and F, the generalized Turán number ex(n, H, F) is the largest number of copies of H in an F -free graph on n vertices. We consider this problem when both H and F have at most
A survey on spectral conditions for some extremal graph problems
This survey is two-fold. We first report new progress on the spectral extremal results on the Tur´an type problems in graph theory. More precisely, we shall summarize the spectral Tur´an function in


The maximum number of cliques in graphs without long cycles
  • Ruth Luo
  • Mathematics
    J. Comb. Theory, Ser. B
  • 2018
Extensions of the Erdős–Gallai theorem and Luo’s theorem
A simple but novel extension of the Erdős–Gallai theorem is established by proving that every graph G contains a path with at least s - 1 edges, where Nj(G) denotes the number of j-cliques in G for 1≤ j ≤ ω(G).
A method in graph theory
A Generalized Turan Problem and its Applications
The problem of how many copies of the k-cycle guarantee the appearance of an ℓ-cycle is considered, and it is shown that for every super-polynomial function $f(\varepsilon )$, there is a family of graphs such that the bounds for the removal lemma are precisely given by f.
Many T copies in H-free graphs
On the size of graphs with complete-factors
The minimum size for any graph to have a Kl-factor is determined and a new short proof of the Erdos-Gallai theorem on the maximum size of graphs with at most β independent edges is given.