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 Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

On strict strong coloring of graphs

2020 ◽  
pp. 2150040
Author(s):  
A. Mohammed Abid ◽  
T. R. Ramesh Rao
Keyword(s):  
Chromatic Number ◽  
Proper Coloring ◽  
Color Class ◽  
Minimum Number ◽  
Strong Chromatic Number

A strict strong coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] in which every vertex of the graph is adjacent to every vertex of some color class. The minimum number of colors required for a strict strong coloring of [Formula: see text] is called the strict strong chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we characterize the results on strict strong coloring of Mycielskian graphs and iterated Mycielskian graphs.

scholarly journals Extremal Bipartite Graphs with Given Parameters on the Resistance–Harary Index

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 615
Author(s):  
Hongzhuan Wang ◽  
Piaoyang Yin
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Extremal Graphs ◽  
Graph Invariant ◽  
Resistance Distance ◽  
Harary Index ◽  
H Index ◽  
Electronic Networks ◽  
Minimum Number ◽  
Graph Parameters

Resistance distance is a concept developed from electronic networks. The calculation of resistance distance in various circuits has attracted the attention of many engineers. This report considers the resistance-based graph invariant, the Resistance–Harary index, which represents the sum of the reciprocal resistances of any vertex pair in the figure G, denoted by R H ( G ) . Vertex bipartiteness in a graph G is the minimum number of vertices removed that makes the graph G become a bipartite graph. In this study, we give the upper bound and lower bound of the R H index, and describe the corresponding extremal graphs in the bipartite graph of a given order. We also describe the graphs with maximum R H index in terms of graph parameters such as vertex bipartiteness, cut edges, and matching numbers.

BIPARTITE SUBGRAPHS OF -FREE GRAPHS

2017 ◽  
Vol 96 (1) ◽  
pp. 1-13 ◽  
Author(s):  
QINGHOU ZENG ◽  
JIANFENG HOU
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Maximum Degree ◽  
Bipartite Subgraph

For a graph $G$, let $f(G)$ denote the maximum number of edges in a bipartite subgraph of $G$. For an integer $m$ and for a fixed graph $H$, 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 $H$. We give a general lower bound for $f(m,H)$ which extends a result of Erdős and Lovász and we study this function for any bipartite graph $H$ with maximum degree at most $t\geq 2$ on one side.

scholarly journals Introduction to dominated edge chromatic number of a graph

2021 ◽  
Vol 41 (2) ◽  
pp. 245-257
Author(s):  
Mohammad R. Piri ◽  
Saeid Alikhani
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Color Class ◽  
Proper Edge Coloring ◽  
Minimum Number

We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph \(G\), is a proper edge coloring of \(G\) such that each color class is dominated by at least one edge of \(G\). The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by \(\chi_{dom}^{\prime}(G)\). We obtain some properties of \(\chi_{dom}^{\prime}(G)\) and compute it for specific graphs. Also examine the effects on \(\chi_{dom}^{\prime}(G)\), when \(G\) is modified by operations on vertex and edge of \(G\). Finally, we consider the \(k\)-subdivision of \(G\) and study the dominated edge chromatic number of these kind of graphs.

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.

THE LINEAR 6-ARBORICITY OF THE COMPLETE BIPARTITE GRAPH Km,n

2013 ◽  
Vol 05 (04) ◽  
pp. 1350029
Author(s):  
SHENGJIE HE ◽  
LIANCUI ZUO
Keyword(s):  
Bipartite Graph ◽  
Undirected Graph ◽  
Complete Bipartite Graph ◽  
Linear Arboricity ◽  
Minimum Number

A linear k-forest of an undirected graph G is a subgraph of G whose components are paths with lengths at most k. The linear k-arboricity of G, denote by lak(G), is the minimum number of linear k-forests needed to partition the edge set E(G) of G. In the case where the lengths of paths are not restricted, we then have the linear arboricity of G which is denoted by la(G). In this paper, we obtain some results about the linear 6-arboricity of the complete bipartite graph Km,n.

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.

b-Chromatic sum of a graph

2015 ◽  
Vol 07 (04) ◽  
pp. 1550040 ◽  
Author(s):  
P. C. Lisna ◽  
M. S. Sunitha
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Double Star ◽  
Star Graph ◽  
Proper Coloring ◽  
Color Class ◽  
Wheel Graph ◽  
Chromatic Sum

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by [Formula: see text], is the maximum integer [Formula: see text] such that G admits a b-coloring with [Formula: see text] colors. In this paper we introduce a new concept, the b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text] and is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for all [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. Also obtained the b-chromatic sum of paths, cycles, wheel graph, complete graph, star graph, double star graph, complete bipartite graph, corona of paths and corona of cycles.

Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions

2003 ◽  
Vol 12 (5-6) ◽  
pp. 477-494 ◽  
Author(s):  
Noga Alon ◽  
Michael Krivelevich ◽  
Benny Sudakov
Keyword(s):  
Positive Constant ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Ramsey Number ◽  
Absolute Positive Constant ◽  
Turán Number ◽  
Minimum Number ◽  
Ramsey Type ◽  
General Graphs ◽  
Refined Version

For a graph H and an integer n, the Turán number is the maximum possible number of edges in a simple graph on n vertices that contains no copy of H. H is r-degenerate if every one of its subgraphs contains a vertex of degree at most r. We prove that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, . This is tight for all values of r and can also be derived from an earlier result of Füredi. We also show that there is an absolute positive constant c such that, for every fixed bipartite r-degenerate graph H, This is motivated by a conjecture of Erdős that asserts that, for every such H, For two graphs G and H, the Ramsey number is the minimum number n such that, in any colouring of the edges of the complete graph on n vertices by red and blue, there is either a red copy of G or a blue copy of H. Erdős conjectured that there is an absolute constant c such that, for any graph G with m edges, . Here we prove this conjecture for bipartite graphs G, and prove that for general graphs G with m edges, for some absolute positive constant c.These results and some related ones are derived from a simple and yet surprisingly powerful lemma, proved, using probabilistic techniques, at the beginning of the paper. This lemma is a refined version of earlier results proved and applied by various researchers including Rödl, Kostochka, Gowers and Sudakov.

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