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.

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 Fractional Biclique Covers and Partitions of Graphs

10.37236/1100 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Valerie L. Watts
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Vertex Cover ◽  
Binary Matrix ◽  
Partition Number ◽  
Bipartite Subgraph ◽  
The Matrix ◽  
Boolean Rank ◽  
Complete Bipartite Subgraph

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.

MAXIMUM CUTS IN GRAPHS WITHOUT WHEELS

2018 ◽  
Vol 100 (1) ◽  
pp. 13-26
Author(s):  
JING LIN ◽  
QINGHOU ZENG ◽  
FUYUAN CHEN
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Bipartite Subgraph ◽  
Wheel Graph

For a graph $G$, let $f(G)$ denote the maximum number of edges in a bipartite subgraph of $G$. Given a fixed graph $H$ and a positive integer $m$, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$, as $G$ ranges over all graphs on $m$ edges that contain no copy of $H$. Alon et al. [‘Maximum cuts and judicious partitions in graphs without short cycles’, J. Combin. Theory Ser. B 88 (2003), 329–346] conjectured that, for any fixed graph $H$, there exists an $\unicode[STIX]{x1D716}(H)>0$ such that $f(m,H)\geq m/2+\unicode[STIX]{x1D6FA}(m^{3/4+\unicode[STIX]{x1D716}})$. We show that, for any wheel graph $W_{2k}$ of $2k$ spokes, there exists $c(k)>0$ such that $f(m,W_{2k})\geq m/2+c(k)m^{(2k-1)/(3k-1)}\log m$. In particular, we confirm the conjecture asymptotically for $W_{4}$ and give general lower bounds for $W_{2k+1}$.

2-Distance chromatic number of some graph products

2020 ◽  
Vol 12 (02) ◽  
pp. 2050021
Author(s):  
Ghazale Ghazi ◽  
Freydoon Rahbarnia ◽  
Mostafa Tavakoli
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Graph Products ◽  
Lexicographic Product ◽  
Graph Product ◽  
Minimum Number

This paper studies the 2-distance chromatic number of some graph product. A coloring of [Formula: see text] is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of [Formula: see text] is the 2-distance chromatic number and denoted by [Formula: see text]. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.

scholarly journals On the Chromatic Thresholds of Hypergraphs

2015 ◽  
Vol 25 (2) ◽  
pp. 172-212
Author(s):  
JÓZSEF BALOGH ◽  
JANE BUTTERFIELD ◽  
PING HU ◽  
JOHN LENZ ◽  
DHRUV MUBAYI
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Open Problems ◽  
Fibre Bundles ◽  
Uniform Hypergraphs ◽  
The Family ◽  
Turán Number ◽  
Chromatic Thresholds ◽  
Extremal Set

Let $\mathcal{F}$ be a family of r-uniform hypergraphs. The chromatic threshold of $\mathcal{F}$ is the infimum of all non-negative reals c such that the subfamily of $\mathcal{F}$ comprising hypergraphs H with minimum degree at least $c \binom{| V(H) |}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs.Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of non-degenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle.In order to prove upper bounds we introduce the concept of fibre bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fibre bundle dimension, a structural property of fibre bundles that is based on the idea of Vapnik–Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

scholarly journals A Note on Embedding Hypertrees

10.37236/256 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Po-Shen Loh
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Classical Result ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs

A classical result from graph theory is that every graph with chromatic number $\chi > t$ contains a subgraph with all degrees at least $t$, and therefore contains a copy of every $t$-edge tree. Bohman, Frieze, and Mubayi recently posed this problem for $r$-uniform hypergraphs. An $r$-tree is a connected $r$-uniform hypergraph with no pair of edges intersecting in more than one vertex, and no sequence of distinct vertices and edges $(v_1, e_1, \ldots, v_k, e_k)$ with all $e_i \ni \{v_i, v_{i+1}\}$, where we take $v_{k+1}$ to be $v_1$. Bohman, Frieze, and Mubayi proved that $\chi > 2rt$ is sufficient to embed every $r$-tree with $t$ edges, and asked whether the dependence on $r$ was necessary. In this note, we completely solve their problem, proving the tight result that $\chi > t$ is sufficient to embed any $r$-tree with $t$ edges.

scholarly journals Spectral Lower Bounds for the Quantum Chromatic Number of a Graph – Part II

10.37236/9295 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Pawel Wocjan ◽  
Clive Elphick ◽  
Parisa Darbari
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Kneser Graph

Hoffman proved that a graph $G$ with eigenvalues $\mu_1 \geqslant \cdots \geqslant \mu_n$ and chromatic number $\chi(G)$ satisfies: \[\chi \geqslant 1 + \kappa\] where $\kappa$ is the smallest integer such that \[\mu_1 + \sum_{i=1}^{\kappa} \mu_{n+1-i} \leqslant 0.\] We strengthen this well known result by proving that $\chi(G)$ can be replaced by the quantum chromatic number, $\chi_q(G)$, where for all graphs $\chi_q(G) \leqslant \chi(G)$ and for some graphs $\chi_q(G)$ is significantly smaller than $\chi(G)$. We also prove a similar result, and investigate implications of these inequalities for the quantum chromatic number of various classes of graphs, which improves many known results. For example, we demonstrate that the Kneser graph $KG_{p,2}$ has $\chi_q = \chi = p - 2$.

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.

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