More Like this

1. Making K _r+1 -free graphs r -partite

   https://doi.org/10.1017/S0963548320000590  

   Balogh, József; Clemen, Felix Christian; Lavrov, Mikhail; Lidický, Bernard; Pfender, Florian (July 2021, Combinatorics, Probability and Computing)

   null (Ed.)

   Abstract The Erdős–Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a K r+ 1 -free graph on n vertices with e ( G ) > ex( n , K r +1 )– α n 2 , then one can remove εn 2 edges from G to obtain an r -partite graph. Füredi gave a short proof that one can choose α = ε . We give a bound for the relationship of α and ε which is asymptotically sharp as ε → 0. 

   more » « less
2. On Turán exponents of bipartite graphs

   https://doi.org/10.1017/S0963548321000341  

   Jiang, Tao; Ma, Jie; Yepremyan, Liana (March 2022, Combinatorics, Probability and Computing)

   Abstract A long-standing conjecture of Erdős and Simonovits asserts that for every rational number $$r\in (1,2)$$ there exists a bipartite graph H such that $$\mathrm{ex}(n,H)=\Theta(n^r)$$ . So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$ , for integers $$k\geq 2$$ . In this paper, we add a new form of rationals for which the conjecture is true: $2-2/(2k+1)$ , for $$k\geq 2$$ . This in turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erdős and Simonovits $$^{\prime}$$ s conjecture, where one replaces a single graph by a finite family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erdős and Simonovits $$^{\prime}$$ s conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon $$^{\prime}$$ s conjecture. We also prove an upper bound on the Turán number of theta graphs in an asymmetric setting and employ this result to obtain another new rational exponent for Turán exponents: $r=7/5$ . 

   more » « less
3. Codegree Conditions for Tiling Complete k-Partite k-Graphs and Loose Cycles

   https://doi.org/10.1017/S096354831900021X  

   Gao, Wei; Han, Jie; Zhao, Yi (November 2019, Combinatorics, Probability and Computing)

   Abstract Given two k -graphs ( k -uniform hypergraphs) F and H , a perfect F -tiling (or F -factor) in H is a set of vertex-disjoint copies of F that together cover the vertex set of H . For all complete k -partite k -graphs K , Mycroft proved a minimum codegree condition that guarantees a K -factor in an n -vertex k -graph, which is tight up to an error term o ( n ). In this paper we improve the error term in Mycroft’s result to a sublinear term that relates to the Turán number of K when the differences of the sizes of the vertex classes of K are co-prime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases, thus disproving a conjecture of Mycroft. Finally, we determine exact minimum codegree conditions for tiling K (k) (1, … , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively. 

   more » « less
4. Sharp bounds for decomposing graphs into edges and triangles

   https://doi.org/10.1017/S0963548320000358  

   Blumenthal, Adam; Lidický, Bernard; Pehova, Yanitsa; Pfender, Florian; Pikhurko, Oleg; Volec, Jan (March 2021, Combinatorics, Probability and Computing)

   null (Ed.)

   Abstract For a real constant α , let $$\pi _3^\alpha (G)$$ be the minimum of twice the number of K 2 ’s plus α times the number of K 3 ’s over all edge decompositions of G into copies of K 2 and K 3 , where K r denotes the complete graph on r vertices. Let $$\pi _3^\alpha (n)$$ be the maximum of $$\pi _3^\alpha (G)$$ over all graphs G with n vertices. The extremal function $$\pi _3^3(n)$$ was first studied by Győri and Tuza ( Studia Sci. Math. Hungar. 22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova ( Combin. Probab. Comput. 28 (2019) 465–472) proved via flag algebras that $$\pi _3^3(n) \le (1/2 + o(1)){n^2}$$ . We extend their result by determining the exact value of $$\pi _3^\alpha (n)$$ and the set of extremal graphs for all α and sufficiently large n . In particular, we show for α = 3 that K n and the complete bipartite graph $${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$$ are the only possible extremal examples for large n . 

   more » « less
5. Tight paths in convex geometric hypergraphs

   https://doi.org/10.19086/aic.12044  

   Mubayi, Dhruv; Füredi, Zoltán; Verstraëte, Jacques; Kostochka, Alexandr; Jiang, Tao (February 2020, Advances in Combinatorics)

   One of the most intruguing conjectures in extremal graph theory is the conjecture of Erdős and Sós from 1962, which asserts that every $$n$$-vertex graph with more than $$\frac{k-1}{2}n$$ edges contains any $$k$$-edge tree as a subgraph. Kalai proposed a generalization of this conjecture to hypergraphs. To explain the generalization, we need to define the concept of a tight tree in an $$r$$-uniform hypergraph, i.e., a hypergraph where each edge contains $$r$$ vertices. A tight tree is an $$r$$-uniform hypergraph such that there is an ordering $$v_1,\ldots,v_n$$ of its its vertices with the following property: the vertices $$v_1,\ldots,v_r$$ form an edge and for every $i>r$, there is a single edge $$e$$ containing the vertex $$v_i$$ and $r-1$ of the vertices $$v_1,\ldots,v_{i-1}$$, and $$e\setminus\{v_i\}$$ is a subset of one of the edges consisting only of vertices from $$v_1,\ldots,v_{i-1}$$. The conjecture of Kalai asserts that every $$n$$-vertex $$r$$-uniform hypergraph with more than $$\frac{k-1}{r}\binom{n}{r-1}$$ edges contains every $$k$$-edge tight tree as a subhypergraph. The recent breakthrough results on the existence of combinatorial designs by Keevash and by Glock, Kühn, Lo and Osthus show that this conjecture, if true, would be tight for infinitely many values of $$n$$ for every $$r$$ and $$k$$.The article deals with the special case of the conjecture when the sought tight tree is a path, i.e., the edges are the $$r$$-tuples of consecutive vertices in the above ordering. The case $r=2$ is the famous Erdős-Gallai theorem on the existence of paths in graphs. The case $r=3$ and $k=4$ follows from an earlier work of the authors on the conjecture of Kalai. The main result of the article is the first non-trivial upper bound valid for all $$r$$ and $$k$$. The proof is based on techniques developed for a closely related problem where a hypergraph comes with a geometric structure: the vertices are points in the plane in a strictly convex position and the sought path has to zigzag beetwen the vertices. 

   more » « less