No Arabic abstract
Let $rge 3$. Given an $r$-graph $H$, the minimum codegree $delta_{r-1}(H)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(H)$ is contained in at least $t$ edges of $H$. Given an $r$-graph $F$, the codegree Turan density $gamma(F)$ is the smallest $gamma >0$ such that every $r$-graph on $n$ vertices with $delta_{r-1}(H)ge (gamma + o(1))n$ contains $F$ as a subhypergraph. Using results on the independence number of hypergraphs, we show that there are constants $c_1, c_2>0$ depending only on $r$ such that [ 1 - c_2 frac{ln t}{t^{r-1}} le gamma(K_t^r) le 1 - c_1 frac{ln t}{t^{r-1}}, ] where $K_t^r$ is the complete $r$-graph on $t$ vertices. This gives the best general bounds for $gamma(K_t^r)$.
Determine the size of $r$-graphs with given graph parameters is an interesting problem. Chvatal and Hanson (JCTB, 1976) gave a tight upper bound of the size of 2-graphs with restricted maximum degree and matching number; Khare (DM, 2014) studied the same problem for linear $3$-graphs with restricted matching number and maximum degree. In this paper, we give a tight upper bound of the size of $3$-graphs with bounded codegree and matching number.
A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in [7]. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It has been also applied in spectral graph theory. Estimating the Lagrangians of hypergraphs has been successfully applied in the course of studying the Turan densities of several hypergraphs as well. It is useful in practice if Motzkin-Straus type results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus result to hypergraphs is false. We attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. In this paper, we give some Motzkin-Straus type results for r-uniform hypergraphs. These results generalize and refine a result of Talbot in [19] and a result in [11].
Let $mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $mathrm{rex}(n, F)$, that are best possible up to a constant factor, when $F$ is one of $C_4$, $K_{2,t}$, $K_{3,3}$ or $K_{s,t}$ when $t>s!$.
We show that for every {eta} > 0 there exists an integer n_0 such that every 2-colouring of the 3-uniform complete hypergraph on n geq n_0 vertices contains two disjoint monochromatic tight cycles of distinct colours that together cover all but at most {eta}n vertices. The same result holds if we replace tight cycles with loose cycles.
Let $kge 3$ be an odd integer and let $n$ be a sufficiently large integer. We prove that the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $2$-regular subgraphs is $binom{n-1}{k-1} + lfloorfrac{n-1}{k} rfloor$, and the equality holds if and only if $H$ is a full $k$-star with center $v$ together with a maximal matching omitting $v$. This verifies a conjecture of Mubayi and Verstra{e}te.