An Upper Bound on Zarankiewicz' Problem

1996 ◽  
Vol 5 (1) ◽  
pp. 29-33 ◽  
Author(s):  
Zoltán Füredi
Keyword(s):  
Upper Bound ◽  
Free Graph ◽  
Asymptotically Optimal ◽  
Zarankiewicz Problem

Let ex(n, K3,3) denote the maximum number of edges of a K3,3-free graph on n vertices. Improving earlier results of Kővári, T. Sós and Turán on Zarankiewicz' problem, we obtain that Brown's example for a maximal K3,3-free graph is asymptotically optimal. Hence .

scholarly journals Some remarks on the Zarankiewicz problem

2021 ◽  
pp. 1-7
Author(s):  
DAVID CONLON
Keyword(s):  
Upper Bound ◽  
Algebraic Method ◽  
Zarankiewicz Problem

Abstract The Zarankiewicz problem asks for an estimate on z(m, n; s, t), the largest number of 1’s in an m × n matrix with all entries 0 or 1 containing no s × t submatrix consisting entirely of 1’s. We show that a classical upper bound for z(m, n; s, t) due to Kővári, Sós and Turán is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method.

An upper bound on the radius of a 3-edge-connected $$C_{4}$$-free graph

2020 ◽  
Author(s):  
Blessings T. Fundikwa ◽  
Jaya P. Mazorodze ◽  
Simon Mukwembi
Keyword(s):  
Upper Bound ◽  
Free Graph

scholarly journals Spectral Extrema for Graphs: The Zarankiewicz Problem

10.37236/212 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
László Babai ◽  
Barry Guiduli
Keyword(s):  
Bipartite Graph ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Complete Bipartite Graph ◽  
Average Degree ◽  
Largest Eigenvalue ◽  
Zarankiewicz Problem ◽  
The Largest Eigenvalue

Let $G$ be a graph on $n$ vertices with spectral radius $\lambda$ (this is the largest eigenvalue of the adjacency matrix of $G$). We show that if $G$ does not contain the complete bipartite graph $K_{t ,s}$ as a subgraph, where $2\le t \le s$, then $$\lambda \le \Big((s-1)^{1/t }+o(1)\Big)n^{1-1/t }$$ for fixed $t$ and $s$ while $n\to\infty$. Asymptotically, this bound matches the Kővári-Turán-Sós upper bound on the average degree of $G$ (the Zarankiewicz problem).

An Upper Bound on the Diameter of a 3-Edge-Connected C4-Free Graph

2020 ◽  
Vol 51 (4) ◽  
pp. 1931-1938
Author(s):  
Blessings T. Fundikwa ◽  
Jaya P. Mazorodze ◽  
Simon Mukwembi
Keyword(s):  
Upper Bound ◽  
Free Graph

Induced Turán Numbers

2017 ◽  
Vol 27 (2) ◽  
pp. 274-288 ◽  
Author(s):  
PO-SHEN LOH ◽  
MICHAEL TAIT ◽  
CRAIG TIMMONS ◽  
RODRIGO M. ZHOU
Keyword(s):  
Upper Bound ◽  
Independent Interest ◽  
Forbidden Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Vertex Graph ◽  
Recent Theorem ◽  
Fixed Order ◽  
Turán Theorem ◽  
Asymptotic Upper Bound

The classical Kővári–Sós–Turán theorem states that ifGis ann-vertex graph with no copy ofKs,tas a subgraph, then the number of edges inGis at mostO(n2−1/s). We prove that if one forbidsKs,tas aninducedsubgraph, and also forbidsanyfixed graphHas a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a non-trivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a non-trivial upper bound on the number of cliques of fixed order in aKr-free graph with no induced copy ofKs,t. This result is an induced analogue of a recent theorem of Alon and Shikhelman and is of independent interest.

scholarly journals An Upper Bound on the Radius of a 3-Vertex-Connected C4-Free Graph

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Blessings T. Fundikwa ◽  
Jaya P. Mazorodze ◽  
Simon Mukwembi
Keyword(s):  
Upper Bound ◽  
Free Graph

We show that if G is a 3-vertex-connected C4-free graph of order n and radius r, then the inequality r≤2n/9+O1 holds. Moreover, graphs are constructed to show that the bounds are asymptotically sharp.

scholarly journals On Winning Fast in Avoider-Enforcer Games

10.37236/328 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
János Barát ◽  
Miloš Stojaković
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Additive Constant ◽  
Main Interest ◽  
Free Graph ◽  
Positional Games

We analyze the duration of the unbiased Avoider-Enforcer game for three basic positional games. All the games are played on the edges of the complete graph on $n$ vertices, and Avoider's goal is to keep his graph outerplanar, diamond-free and $k$-degenerate, respectively. It is clear that all three games are Enforcer's wins, and our main interest lies in determining the largest number of moves Avoider can play before losing. Extremal graph theory offers a general upper bound for the number of Avoider's moves. As it turns out, for all three games we manage to obtain a lower bound that is just an additive constant away from that upper bound. In particular, we exhibit a strategy for Avoider to keep his graph outerplanar for at least $2n-8$ moves, being just $6$ short of the maximum possible. A diamond-free graph can have at most $d(n)=\lceil\frac{3n-4}{2}\rceil$ edges, and we prove that Avoider can play for at least $d(n)-3$ moves. Finally, if $k$ is small compared to $n$, we show that Avoider can keep his graph $k$-degenerate for as many as $e(n)$ moves, where $e(n)$ is the maximum number of edges a $k$-degenerate graph can have.

scholarly journals Maximum difference about the size of optimal identifying codes in graphs differing by one vertex

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mikko Pelto
Keyword(s):  
Upper Bound ◽  
Undirected Graph ◽  
Minimum Size ◽  
Maximum Difference ◽  
Free Graph ◽  
Induced Subgraph ◽  
Identifying Code ◽  
Vertex Set ◽  
International Audience ◽  
Codes In Graphs

Graph Theory International audience Let G=(V,E) be a simple undirected graph. We call any subset C⊆V an identifying code if the sets I(v)={c∈C | {v,c}∈E or v=c } are distinct and non-empty for all vertices v∈V. A graph is called twin-free if there is an identifying code in the graph. The identifying code with minimum size in a twin-free graph G is called the optimal identifying code and the size of such a code is denoted by γ(G). Let GS denote the induced subgraph of G where the vertex set S⊂V is deleted. We provide a tight upper bound for γ(GS)-γ(G) when both graphs are twin-free and |V| is large enough with respect to |S|. Moreover, we prove tight upper bound when G is a bipartite graph and |S|=1.

Size, Order, and Connected Domination

2014 ◽  
Vol 57 (1) ◽  
pp. 141-144 ◽  
Author(s):  
Simon Mukwembi
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Leaf Number ◽  
Classical Theorem ◽  
Local Independence ◽  
Free Graph

AbstractWe give a sharp upper bound on the size of a triangle-free graph of a given order and connected domination. Our bound, apart from strengthening an old classical theorem of Mantel and of Turén improves on a theorem of Sanchis. Further, as corollaries, we settle a long standing conjecture of Graffiti on the leaf number and local independence for triangle-free graphs and answer a question of Griggs, Kleitman, and Shastri on a lower bound of the leaf number in triangle-free graphs.

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