The Largest Complete Bipartite Subgraph in Point-Hyperplane Incidence Graphs

Given $m$ points and $n$ hyperplanes in $\mathbb{R}^d$ ($d\geqslant 3)$, if there are many incidences, we expect to find a big cluster $K_{r,s}$ in their incidence graph. Apfelbaum and Sharir found lower and upper bounds for the largest size of $rs$, which match (up to a constant) only in three dimensions. In this paper we close the gap in four and five dimensions, up to some polylogarithmic factors.

Colorful Subhypergraphs in Uniform Hypergraphs

There are several topological results ensuring in any properly colored graph the existence of a colorful complete bipartite subgraph, whose order is bounded from below by some topological invariants of some topological spaces associated to the graph. Meunier [Colorful subhypergraphs in Kneser hypergraphs, The Electronic Journal of Combinatorics, 2014] presented the first colorful type result for uniform hypergraphs. In this paper, we give some new generalizations of the $\mathbb{Z}_p$-Tucker lemma and by use of them, we improve Meunier's result and some other colorful results by Simonyi, Tardif, and Zsbán [Colourful theorems and indices of homomorphism complexes, The Electronic Journal of Combinatorics, 2014] and by Simonyi and Tardos [Colorful subgraphs in Kneser-like graphs, European Journal of Combinatorics, 2007] to uniform hypergraphs. Also, we introduce some new lower bounds for the chromatic number and local chromatic number of uniform hypergraphs. A hierarchy between these lower bounds is presented as well.

Polyhedral Graphs of GRAPH PARTITIONING and COMPLETE BIPARTITE SUBGRAPH Problems

We provide an effective description of graphs of polyhedra for GRAPH PARTITIONING and COMPLETE BIPARTITE SUBGRAPH problems. We establish the fact, that the clique number for each of this problems increases exponentially with the dimension of the space.

Colourful Theorems and Indices of Homomorphism Complexes

We extend the colourful complete bipartite subgraph theorems of [G. Simonyi, G. Tardos, Local chromatic number, Ky Fan's theorem, and circular colorings, Combinatorica 26 (2006), 587--626] and [G. Simonyi, G. Tardos, Colorful subgraphs of Kneser-like graphs, European J. Combin. 28 (2007), 2188--2200] to more general topological settings. We give examples showing that the hypotheses are indeed more general. We use our results to show that the topological bounds on chromatic numbers of digraphs with tree duality are at most one better than the clique number. We investigate combinatorial and complexity-theoretic aspects of relevant order-theoretic maps.

A Separator Theorem for String Graphs and its Applications

A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with m edges can be separated into two parts of roughly equal size by the removal of $O(m^{3/4}\sqrt{\log m})$ vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph Kt,t has at most ctn edges, where ct is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any ε > 0, there is an integer g(ε) such that every string graph with n vertices and girth at least g(ε) has at most (1 + ε)n edges. Furthermore, the number of such labelled graphs is at most (1 + ε)nT(n), where T(n) = nn−2 is the number of labelled trees on n vertices.

Fractional Biclique Covers and Partitions of Graphs

A biclique is a complete bipartite subgraph of a graph. This paper investigates the fractional biclique cover number, $bc^*(G)$, and the fractional biclique partition number, $bp^*(G)$, of a graph $G$. It is observed that $bc^*(G)$ and $bp^*(G)$ provide lower bounds on the biclique cover and partition numbers respectively, and conditions for equality are given. It is also shown that $bc^*(G)$ is a better lower bound on the Boolean rank of a binary matrix than the maximum number of isolated ones of the matrix. In addition, it is noted that $bc^*(G) \leq bp^*(G) \leq \beta^*(G)$, the fractional vertex cover number. Finally, the application of $bc^*(G)$ and $bp^*(G)$ to two different weak products is discussed.

On Sampling Colorings of Bipartite Graphs

International audience We study the problem of efficiently sampling k-colorings of bipartite graphs. We show that a class of markov chains cannot be used as efficient samplers. Precisely, we show that, for any k, 6 ≤ k ≤ n^\1/3-ε \, ε > 0 fixed, \emphalmost every bipartite graph on n+n vertices is such that the mixing time of any markov chain asymptotically uniform on its k-colorings is exponential in n/k^2 (if it is allowed to only change the colors of O(n/k) vertices in a single transition step). This kind of exponential time mixing is called \emphtorpid mixing. As a corollary, we show that there are (for every n) bipartite graphs on 2n vertices with Δ (G) = Ω (\ln n) such that for every k, 6 ≤ k ≤ Δ /(6 \ln Δ ), each member of a large class of chains mixes torpidly. While, for fixed k, such negative results are implied by the work of CDF, our results are more general in that they allow k to grow with n. We also show that these negative results hold true for H-colorings of bipartite graphs provided H contains a spanning complete bipartite subgraph. We also present explicit examples of colorings (k-colorings or H-colorings) which admit 1-cautious chains that are ergodic and are shown to have exponential mixing time. While, for fixed k or fixed H, such negative results are implied by the work of CDF, our results are more general in that they allow k or H to vary with n.

Threshold Functions for the Bipartite Turán Property

Let $G_2(n)$ denote a bipartite graph with $n$ vertices in each color class, and let $z(n,t)$ be the bipartite Turán number, representing the maximum possible number of edges in $G_2(n)$ if it does not contain a copy of the complete bipartite subgraph $K(t,t)$. It is then clear that $\zeta(n,t)=n^2-z(n,t)$ denotes the minimum number of zeros in an $n\times n$ zero-one matrix that does not contain a $t\times t$ submatrix consisting of all ones. We are interested in the behaviour of $z(n,t)$ when both $t$ and $n$ go to infinity. The case $2\le t\ll n^{1/5}$ has been treated elsewhere; here we use a different method to consider the overlapping case $\log n\ll t\ll n^{1/3}$. Fill an $n \times n$ matrix randomly with $z$ ones and $\zeta=n^2-z$ zeros. Then, we prove that the asymptotic probability that there are no $t \times t$ submatrices with all ones is zero or one, according as $z\ge(t/ne)^{2/t}\exp\{a_n/t^2\}$ or $z\le(t/ne)^{2/t}\exp\{(\log t-b_n)/t^2\}$, where $a_n$ tends to infinity at a specified rate, and $b_n\to\infty$ is arbitrary. The proof employs the extended Janson exponential inequalities.