scholarly journals A Separator Theorem for String Graphs and its Applications

2009 ◽  
Vol 19 (3) ◽  
pp. 371-390 ◽  
Author(s):  
JACOB FOX ◽  
JÁNOS PACH
Keyword(s):  
Intersection Graph ◽  
Equal Size ◽  
Bipartite Subgraph ◽  
String Graphs ◽  
String Graph ◽  
Complete Bipartite Subgraph ◽  
Labelled Trees ◽  
Separator Theorem ◽  
Labelled Graphs

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.

scholarly journals Applications of a New Separator Theorem for String Graphs

2013 ◽  
Vol 23 (1) ◽  
pp. 66-74 ◽  
Author(s):  
JACOB FOX ◽  
JÁNOS PACH
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Present Note ◽  
Intersection Graph ◽  
Type Result ◽  
Vertex Separator ◽  
String Graphs ◽  
String Graph ◽  
Ramsey Type ◽  
Separator Theorem

An intersection graph of curves in the plane is called astring graph. Matoušek almost completely settled a conjecture of the authors by showing that every string graph withmedges admits a vertex separator of size$O(\sqrt{m}\log m)$. In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphsGwithnvertices: (i) ifKt⊈Gfor somet, then the chromatic number ofGis at most (logn)O(logt); (ii) ifKt,t⊈G, thenGhas at mostt(logt)O(1)nedges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdős–Hajnal conjecture almost holds for string graphs.

scholarly journals Near-Optimal Separators in String Graphs

2013 ◽  
Vol 23 (1) ◽  
pp. 135-139 ◽  
Author(s):  
JIŘÍ MATOUŠEK
Keyword(s):  
Intersection Graph ◽  
String Graphs ◽  
String Graph

Let G be a string graph (an intersection graph of continuous arcs in the plane) with m edges. Fox and Pach proved that G has a separator consisting of $O(m^{3/4}\sqrt{\log m})$ vertices, and they conjectured that the bound of $O(\sqrt m)$ actually holds. We obtain separators with $O(\sqrt m \,\log m)$ vertices.

scholarly journals A Sharp Threshold Phenomenon in String Graphs

2021 ◽  
Author(s):  
István Tomon
Keyword(s):  
Recent Result ◽  
Direct Consequence ◽  
Intersection Graph ◽  
Edge Density ◽  
Sharp Threshold ◽  
Threshold Phenomenon ◽  
String Graphs ◽  
String Graph ◽  
Special Case

AbstractA string graph is the intersection graph of curves in the plane. We prove that for every $$\epsilon >0$$ ϵ > 0 , if G is a string graph with n vertices such that the edge density of G is below $${1}/{4}-\epsilon $$ 1 / 4 - ϵ , then V(G) contains two linear sized subsets A and B with no edges between them. The constant 1/4 is a sharp threshold for this phenomenon as there are string graphs with edge density less than $${1}/{4}+\epsilon $$ 1 / 4 + ϵ such that there is an edge connecting any two logarithmic sized subsets of the vertices. The existence of linear sized sets A and B with no edges between them in sufficiently sparse string graphs is a direct consequence of a recent result of Lee about separators. Our main theorem finds the largest possible density for which this still holds. In the special case when the curves are x-monotone, the same result was proved by Pach and the author of this paper, who also proposed the conjecture for the general case.

scholarly journals Threshold Functions for the Bipartite Turán Property

10.37236/1303 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Anant P. Godbole ◽  
Ben Lamorte ◽  
Erik Jonathan Sandquist
Keyword(s):  
Bipartite Graph ◽  
Exponential Inequalities ◽  
Threshold Functions ◽  
Number Of Zeros ◽  
Bipartite Subgraph ◽  
Color Class ◽  
Turán Number ◽  
Minimum Number ◽  
Complete Bipartite Subgraph ◽  
Asymptotic Probability

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.

scholarly journals Colourful Theorems and Indices of Homomorphism Complexes

10.37236/2302 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Gábor Simonyi ◽  
Claude Tardif ◽  
Ambrus Zsbán
Keyword(s):  
Chromatic Number ◽  
Clique Number ◽  
Chromatic Numbers ◽  
Bipartite Subgraph ◽  
Complete Bipartite Subgraph ◽  
Better Than

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.

scholarly journals On Sampling Colorings of Bipartite Graphs

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
R. Balasubramanian ◽  
C.R. Subramanian
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Mixing Time ◽  
Bipartite Graphs ◽  
Exponential Time ◽  
Negative Results ◽  
Bipartite Subgraph ◽  
Single Transition ◽  
International Audience ◽  
Complete Bipartite Subgraph

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.

scholarly journals Polyhedral Graphs of GRAPH PARTITIONING and COMPLETE BIPARTITE SUBGRAPH Problems

2015 ◽  
Vol 19 (6) ◽  
pp. 101-106
Author(s):  
A. I. Antonov ◽  
V. A. Bondarenko
Keyword(s):  
Graph Partitioning ◽  
Clique Number ◽  
Bipartite Subgraph ◽  
Complete Bipartite Subgraph ◽  
Effective Description

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.

scholarly journals The Largest Complete Bipartite Subgraph in Point-Hyperplane Incidence Graphs

10.37236/8253 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Thao Do
Keyword(s):  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Incidence Graph ◽  
Five Dimensions ◽  
Bipartite Subgraph ◽  
Complete Bipartite Subgraph

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.

An Introduction to Intersection Graphs

2020 ◽  
pp. 24-65 ◽  
Author(s):  
Madhumangal Pal
Keyword(s):  
Interval Graphs ◽  
Chordal Graphs ◽  
Intersection Graph ◽  
Important Class ◽  
Intersection Graphs ◽  
Permutation Graphs ◽  
Circular Arc ◽  
String Graphs ◽  
Circular Arc Graphs ◽  
Trapezoid Graphs

In this chapter, a very important class of graphs called intersection graph is introduced. Based on the geometrical representation, many different types of intersection graphs can be defined with interesting properties. Some of them—interval graphs, circular-arc graphs, permutation graphs, trapezoid graphs, chordal graphs, line graphs, disk graphs, string graphs—are presented here. A brief introduction of each of these intersection graphs along with some basic properties and algorithmic status are investigated.

scholarly journals Colorful Subhypergraphs in Uniform Hypergraphs

10.37236/6154 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Meysam Alishahi
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Topological Spaces ◽  
Topological Invariants ◽  
Electronic Journal ◽  
Colored Graph ◽  
Bipartite Subgraph ◽  
Uniform Hypergraphs ◽  
Complete Bipartite Subgraph ◽  
Tucker Lemma

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.

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