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1. 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: $r=7/5$ . 

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2. Improved Bounds on a Generalization of Tuza's Conjecture

   https://doi.org/10.37236/10829  

   Basit, Abdul; McGinnis, Daniel; Simmons, Henry; Sinnwell, Matt; Zerbib, Shira (October 2022, The Electronic Journal of Combinatorics)

   For an $$r$$-uniform hypergraph $$H$$, let $$\nu^{(m)}(H)$$ denote the maximum size of a set $$M$$ of edges in $$H$$ such that every two edges in $$M$$ intersect in less than $$m$$ vertices, and let $$\tau^{(m)}(H)$$ denote the minimum size of a collection $$C$$ of $$m$$-sets of vertices such that every edge in $$H$$ contains an element of $$C$$. The fractional analogues of these parameters are denoted by $$\nu^{*(m)}(H)$$ and $$\tau^{*(m)}(H)$$, respectively. Generalizing a famous conjecture of Tuza on covering triangles in a graph, Aharoni and Zerbib conjectured that for every $$r$$-uniform hypergraph $$H$$, $$\tau^{(r-1)}(H)/\nu^{(r-1)}(H) \leq \lceil{\frac{r+1}{2}}\rceil$$. In this paper we prove bounds on the ratio between the parameters $$\tau^{(m)}$$ and $$\nu^{(m)}$$, and their fractional analogues. Our main result is that, for every $$r$$-uniform hypergraph~$$H$$,\[ \tau^{*(r-1)}(H)/\nu^{(r-1)}(H) \le \begin{cases} \frac{3}{4}r - \frac{r}{4(r+1)} &\text{for }r\text{ even,}\\\frac{3}{4}r - \frac{r}{4(r+2)} &\text{for }r\text{ odd.} \\\end{cases} \]This improves the known bound of $$r-1$.We also prove that, for every $$r$$-uniform hypergraph $$H$$, $$\tau^{(m)}(H)/\nu^{*(m)}(H) \le \operatorname{ex}_m(r, m+1)$$, where the Turán number $$\operatorname{ex}_r(n, k)$$ is the maximum number of edges in an $$r$$-uniform hypergraph on $$n$$ vertices that does not contain a copy of the complete $$r$$-uniform hypergraph on $$k$$ vertices. Finally, we prove further bounds in the special cases $(r,m)=(4,2)$ and $(r,m)=(4,3)$. 

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3. A Faster Algorithm for Minimum-cost Bipartite Perfect Matching in Planar Graphs

   https://doi.org/10.1145/3365006  

   Asathulla, Mudabir Kabir; Khanna, Sanjeev; Lahn, Nathaniel; Raghvendra, Sharath (January 2020, ACM Transactions on Algorithms)

   null (Ed.)

   Given a weighted planar bipartite graph G ( A ∪ B , E ) where each edge has an integer edge cost, we give an Õ( n 4/3 log nC ) time algorithm to compute minimum-cost perfect matching; here C is the maximum edge cost in the graph. The previous best-known planarity exploiting algorithm has a running time of O ( n 3/2 log n ) and is achieved by using planar separators (Lipton and Tarjan ’80). Our algorithm is based on the bit-scaling paradigm (Gabow and Tarjan ’89). For each scale, our algorithm first executes O ( n 1/3 ) iterations of Gabow and Tarjan’s algorithm in O ( n 4/3 ) time leaving only O ( n 2/3 ) vertices unmatched. Next, it constructs a compressed residual graph H with O ( n 2/3 ) vertices and O ( n ) edges. This is achieved by using an r -division of the planar graph G with r = n 2/3 . For each partition of the r -division, there is an edge between two vertices of H if and only if they are connected by a directed path inside the partition. Using existing efficient shortest-path data structures, the remaining O ( n 2/3 ) vertices are matched by iteratively computing a minimum-cost augmenting path, each taking Õ( n 2/3 ) time. Augmentation changes the residual graph, so the algorithm updates the compressed representation for each partition affected by the change in Õ( n 2/3 ) time. We bound the total number of affected partitions over all the augmenting paths by O ( n 2/3 log n ). Therefore, the total time taken by the algorithm is Õ( n 4/3 ). 

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4. 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. 

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5. Generalizations and Strengthenings of Ryser's Conjecture

   https://doi.org/10.37236/9914  

   DeBiasio, Louis; Kamel, Yigal; McCourt, Grace; Sheats, Hannah (October 2021, The Electronic Journal of Combinatorics)

   Ryser's conjecture says that for every $$r$$-partite hypergraph $$H$$ with matching number $$\nu(H)$$, the vertex cover number is at most $$(r-1)\nu(H)$$.  This far-reaching generalization of König's theorem is only known to be true for $$r\leq 3$$, or when $$\nu(H)=1$$ and $$r\leq 5$$.  An equivalent formulation of Ryser's conjecture is that in every $$r$$-edge coloring of a graph $$G$$ with independence number $$\alpha(G)$$, there exists at most $$(r-1)\alpha(G)$$ monochromatic connected subgraphs which cover the vertex set of $$G$$.   We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs.  Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results. 

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