scholarly journals Cliques in Graphs Excluding a Complete Graph Minor

10.37236/5715 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
David R. Wood
Keyword(s):  
Complete Graph ◽  
Extremal Function ◽  
Graph Minor ◽  
Exact Answer ◽  
Vertex Graph ◽  
The Extremal Function

This paper considers the following question: What is the maximum number of $k$-cliques in an $n$-vertex graph with no $K_t$-minor? This question generalises the extremal function for $K_t$-minors, which corresponds to the $k=2$ case. The exact answer is given for $t\leq 9$ and all values of $k$. We also determine the maximum total number of cliques in an $n$-vertex graph with no $K_t$-minor for $t\leq 9$. Several observations are made about the case of general $t$.

scholarly journals Forbidden Berge Hypergraphs

10.37236/6482 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
R. P. Anstee ◽  
Santiago Salazar
Keyword(s):  
Extremal Function ◽  
Column Permutation ◽  
Vertex Graph ◽  
Simple Matrix ◽  
The Extremal Function

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say that a (0,1)-matrix $A$ has $F$ as a Berge hypergraph if there is a submatrix $B$ of $A$ and some row and column permutation of $F$, say $G$, with $G\le B$. Letting $\|A\|$ denote the number of columns in $A$, we define the extremal function $\mathrm{Bh}(m,{ F})=\max\{\|A\|\,:\, A \hbox{ }m\hbox{-rowed simple matrix and no Berge hypergraph }F\}$. We determine the asymptotics of $\mathrm{Bh}(m,F)$ for all $3$- and $4$-rowed $F$ and most $5$-rowed $F$. For certain $F$, this becomes the problem of determining the maximum number of copies of $K_r$ in a $m$-vertex graph that has no $K_{s,t}$ subgraph, a problem studied by Alon and Shikhelman.

scholarly journals Sharp bounds for decomposing graphs into edges and triangles

2020 ◽  
pp. 1-17
Author(s):  
Adam Blumenthal ◽  
Bernard Lidický ◽  
Yanitsa Pehova ◽  
Florian Pfender ◽  
Oleg Pikhurko ◽  
...  
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Extremal Function ◽  
Recent Progress ◽  
Complete Bipartite Graph ◽  
Real Constant ◽  
Extremal Graphs ◽  
Sharp Bounds ◽  
Flag Algebras ◽  
The Extremal Function

Abstract For a real constant α, let $\pi _3^\alpha (G)$ be the minimum of twice the number of K2’s plus α times the number of K3’s over all edge decompositions of G into copies of K2 and K3, where Kr denotes the complete graph on r vertices. Let $\pi _3^\alpha (n)$ be the maximum of $\pi _3^\alpha (G)$ over all graphs G with n vertices. The extremal function $\pi _3^3(n)$ was first studied by Győri and Tuza (Studia Sci. Math. Hungar.22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova (Combin. Probab. Comput.28 (2019) 465–472) proved via flag algebras that $\pi _3^3(n) \le (1/2 + o(1)){n^2}$ . We extend their result by determining the exact value of $\pi _3^\alpha (n)$ and the set of extremal graphs for all α and sufficiently large n. In particular, we show for α = 3 that Kn and the complete bipartite graph ${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$ are the only possible extremal examples for large n.

scholarly journals A majorant problem

1992 ◽  
Vol 15 (3) ◽  
pp. 441-447
Author(s):  
Ronen Peretz
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Fourier Coefficients ◽  
Positive Real Part ◽  
Complex Vector ◽  
Positive Real ◽  
Zero Sets ◽  
The Extremal Function

Letf(z)=∑k=0∞akzk,a0≠0be analytic in the unit disc. Any infinite complex vectorθ=(θ0,θ1,θ2,…)such that|θk|=1,k=0,1,2,…, induces a functionfθ(z)=∑k=0∞akθkzkwhich is still analytic in the unit disc.In this paper we study the problem of maximizing thep-means:∫02π|fθ(reiϕ)|pdϕover all possible vectorsθand for values ofrclose to0and for allp<2.It is proved that a maximizing function isf1(z)=−|a0|+∑k=1∞|ak|zkand thatrcould be taken to be any positive number which is smaller than the radius of the largest disc centered at the origin which can be inscribed in the zero sets off1. This problem is originated by a well known majorant problem for Fourier coefficients that was studied by Hardy and Littlewood.One consequence of our paper is that forp<2the extremal function for the Hardy-Littlewood problem should be−|a0|+∑k=1∞|ak|zk.We also give some applications to derive some sharp inequalities for the classes of Schlicht functions and of functions of positive real part.

scholarly journals The extremal function for partial bipartite tilings

2012 ◽  
Vol 33 (5) ◽  
pp. 807-815 ◽  
Author(s):  
Codruţ Grosu ◽  
Jan Hladký
Keyword(s):  
Extremal Function ◽  
The Extremal Function

Unpredictability of the coefficients of m-fold symmetric bi-starlike functions

2014 ◽  
Vol 25 (07) ◽  
pp. 1450064 ◽  
Author(s):  
Samaneh G. Hamidi ◽  
Jay M. Jahangiri
Keyword(s):  
Extremal Function ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Square Root ◽  
Koebe Function ◽  
The Extremal Function

In 1984, Libera and Zlotkiewicz proved that the inverse of the square-root transform of the Koebe function is the extremal function for the inverses of odd univalent functions. The purpose of this paper is to point out that this is not the case for the m-fold symmetric bi-starlike functions by demonstrating the unpredictability of the coefficients of such functions.

scholarly journals Convolutions of prestarlike functions

1983 ◽  
Vol 6 (1) ◽  
pp. 59-68 ◽  
Author(s):  
O. P. Ahuja ◽  
H. Silverman
Keyword(s):  
Unit Disk ◽  
Extremal Function ◽  
Extreme Points ◽  
Distortion Theorems ◽  
The Extremal Function ◽  
Class Of Functions

The convolution of two functionsf(z)=∑n=0∞anznandg(z)=∑n=0∞bnzndefined as(f∗g)(z)=∑n=0∞anbnzn. Forf(z)=z−∑n=2∞anznandg(z)=z/(1−z)2(1−γ), the extremal function for the class of functions starlike of orderγ, we investigate functionsh, whereh(z)=(f∗g)(z), which satisfy the inequality|(zh′/h)−1|/|(zh′/h)+(1-2α)|<β,0≤α<1,0<β≤1for allzin the unit disk. Such functionsfare said to beγ-prestarlike of orderαand typeβ. We characterize this family in terms of its coefficients, and then determine extreme points, distortion theorems, and radii of univalence, starlikeness, and convexity. All results are sharp.

scholarly journals Note on Schiffer’s Variation in the Class of Univalent Functions in the Unit Disc

1968 ◽  
Vol 32 ◽  
pp. 273-276
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Univalent Functions ◽  
The Real ◽  
Maximum Value ◽  
The Extremal Function

Let S denote the class of univalent functions f(z) in the unit disc D: | z | < 1 with the following expansion: (1) f(z) = z + a2z2 + a3z3 + · · · · anzn + · ··.We denote by fn(z) the extremal function in S which gives the maximum value of the real part of an and by Dn the image of D under w = fn(z).

scholarly journals Extremal problems for convex geometric hypergraphs and ordered hypergraphs

2020 ◽  
pp. 1-19
Author(s):  
Zoltán Füredi ◽  
Tao Jiang ◽  
Alexandr Kostochka ◽  
Dhruv Mubayi ◽  
Jacques Verstraëte
Keyword(s):  
Extremal Function ◽  
Combinatorial Geometry ◽  
Extremal Problems ◽  
Geometric Graphs ◽  
Uniform Hypergraphs ◽  
Vertex Set ◽  
Order Of Magnitude ◽  
The Extremal Function ◽  
Geometric Hypergraphs ◽  
Rich History

Abstract An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braı–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting.

Monochromatic paths and cycles in 2-edge-coloured graphs with large minimum degree

2021 ◽  
pp. 1-14
Author(s):  
József Balogh ◽  
Alexandr Kostochka ◽  
Mikhail Lavrov ◽  
Xujun Liu
Keyword(s):  
Complete Graph ◽  
Minimum Degree ◽  
Dense Subgraph ◽  
Degree Condition ◽  
Vertex Graph ◽  
Delta G ◽  
Edge Colouring ◽  
Monochromatic Copy

Abstract A graph G arrows a graph H if in every 2-edge-colouring of G there exists a monochromatic copy of H. Schelp had the idea that if the complete graph $K_n$ arrows a small graph H, then every ‘dense’ subgraph of $K_n$ also arrows H, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large n, if $n = 3t+r$ where $r \in \{0,1,2\}$ and G is an n-vertex graph with $\delta(G) \ge (3n-1)/4$ , then for every 2-edge-colouring of G, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same colour, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the samecolour. Our result is tight in the sense that no longer cycles (of length $>2t+r$ ) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large n, every $(3t-1)$ -vertex graph G with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with 2n vertices. Moreover, it implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every n-vertex graph G with $\delta(G) \ge 3n/4$ , in each 2-edge-colouring of G there exists a monochromatic cycle of length at least $2t+r$ .

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