scholarly journals The Extremal Function for Cycles of Length $\ell$ mod $k$

10.37236/6257 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Benny Sudakov ◽  
Jacques Verstraete
Keyword(s):  
Bipartite Graph ◽  
Extremal Function ◽  
Absolute Constant ◽  
Complete Bipartite Graph ◽  
Short Note ◽  
Upper Bounds ◽  
Average Degree ◽  
Free Graph ◽  
The Extremal Function

Burr and Erdős conjectured that for each $k,\ell \in \mathbb Z^+$ such that $k \mathbb Z + \ell$ contains even integers, there exists $c_k(\ell)$ such that any graph of average degree at least $c_k(\ell)$ contains a cycle of length $\ell$ mod $k$. This conjecture was proved by Bollobás, and many successive improvements of upper bounds on $c_k(\ell)$ appear in the literature. In this short note, for $1 \leq \ell \leq k$, we show that $c_k(\ell)$ is proportional to the largest average degree of a $C_{\ell}$-free graph on $k$ vertices, which determines $c_k(\ell)$ up to an absolute constant. In particular, using known results on Turán numbers for even cycles, we obtain $c_k(\ell) = O(\ell k^{2/\ell})$ for all even $\ell$, which is tight for $\ell \in \{4,6,10\}$. Since the complete bipartite graph $K_{\ell - 1,n - \ell + 1}$ has no cycle of length $2\ell$ mod $k$, it also shows $c_k(\ell) = \Theta(\ell)$ for $\ell = \Omega(\log k)$.

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 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).

scholarly journals Turán Number of an Induced Complete Bipartite Graph Plus an Odd Cycle

2018 ◽  
Vol 28 (2) ◽  
pp. 241-252 ◽  
Author(s):  
BEKA ERGEMLIDZE ◽  
ERVIN GYŐRI ◽  
ABHISHEK METHUKU
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Free Graph ◽  
Odd Cycle ◽  
Turán Number

Let k ⩾ 2 be an integer. We show that if s = 2 and t ⩾ 2, or s = t = 3, then the maximum possible number of edges in a C2k+1-free graph containing no induced copy of Ks,t is asymptotically equal to (t − s + 1)1/s(n/2)2−1/s except when k = s = t = 2.This strengthens a result of Allen, Keevash, Sudakov and Verstraëte [1], and answers a question of Loh, Tait, Timmons and Zhou [14].

Signed domination and Mycielski’s structure in graphs

2020 ◽  
Vol 54 (4) ◽  
pp. 1077-1086
Author(s):  
Arezoo N. Ghameshlou ◽  
Athena Shaminezhad ◽  
Ebrahim Vatandoost ◽  
Abdollah Khodkar
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Minimum Weight ◽  
Domination Number ◽  
Upper Bounds ◽  
Underlying Graph ◽  
Signed Domination ◽  
Signed Domination Number ◽  
Dominating Function ◽  
Domination Numbers

Let G = (V, E) be a graph. The function f : V(G) → {−1, 1} is a signed dominating function if for every vertex v ∈ V(G), ∑x∈NG[v] f(x)≥1. The value of ω(f) = ∑x∈V(G) f(x) is called the weight of f. The signed domination number of G is the minimum weight of a signed dominating function of G. In this paper, we initiate the study of the signed domination numbers of Mycielski graphs and find some upper bounds for this parameter. We also calculate the signed domination number of the Mycielski graph when the underlying graph is a star, a wheel, a fan, a Dutch windmill, a cycle, a path or a complete bipartite graph.

scholarly journals Forbidden formations in 0-1 matrices

2018 ◽  
Author(s):  
Jesse Geneson
Keyword(s):  
Extremal Function ◽  
Upper Bounds ◽  
Extremal Functions ◽  
New Family ◽  
Permutation Matrices ◽  
Multidimensional Matrices ◽  
The Extremal Function

Keszegh (2009) proved that the extremal function $ex(n, P)$ of any forbidden light $2$-dimensional 0-1 matrix $P$ is at most quasilinear in $n$, using a reduction to generalized Davenport-Schinzel sequences. We extend this result to multidimensional matrices by proving that any light $d$-dimensional 0-1 matrix $P$ has extremal function $ex(n, P,d) = O(n^{d-1}2^{\alpha(n)^{t}})$ for some constant $t$ that depends on $P$. To prove this result, we introduce a new family of patterns called $(P, s)$-formations, which are a generalization of $(r, s)$-formations, and we prove upper bounds on their extremal functions. In many cases, including permutation matrices $P$ with at least two ones, we are able to show that our $(P, s)$-formation upper bounds are tight.

scholarly journals Maximizing the Number of Independent Sets of a Fixed Size

2014 ◽  
Vol 24 (3) ◽  
pp. 521-527 ◽  
Author(s):  
WENYING GAN ◽  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Complete Bipartite Graph ◽  
Short Note ◽  
Independent Sets ◽  
Fixed Size

Letit(G) be the number of independent sets of sizetin a graphG. Engbers and Galvin asked how largeit(G) could be in graphs with minimum degree at least δ. They further conjectured that whenn⩾ 2δ andt⩾ 3,it(G) is maximized by the complete bipartite graphKδ,n−δ. This conjecture has recently drawn the attention of many researchers. In this short note, we prove this conjecture.

scholarly journals Multicolour Bipartite Ramsey Number of Paths

10.37236/8458 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Matija Bucic ◽  
Shoham Letzter ◽  
Benny Sudakov
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Upper Bounds ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Over 40 Years ◽  
Monochromatic Copy

The k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $N$ for which every $k$-edge-coloured complete bipartite graph $K_{N,N}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was initiated over 40 years ago by Faudree and Schelp and, independently, by Gyárfás and Lehel, who determined the $2$-colour bipartite Ramsey number of paths. Recently the $3$-colour Ramsey number of paths and (even) cycles, was essentially determined as well. Improving the results of DeBiasio, Gyárfás, Krueger, Ruszinkó, and Sárközy, in this paper we determine asymptotically the $4$-colour bipartite Ramsey number of paths and cycles. We also provide new upper bounds on the $k$-colour bipartite Ramsey numbers of paths and cycles which are close to being tight.

scholarly journals A class of zero divisor rings in which every graph is precisely the union of a complete graph and a complete bipartite graph

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Syed Khalid Nauman ◽  
Basmah H. Shafee
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Zero Divisor ◽  
Complete Bipartite Graph ◽  
Upper Bounds ◽  
Local Rings ◽  
Lower And Upper Bounds ◽  
Zero Divisor Graph ◽  
Characteristic Two

AbstractRecently, an interest is developed in estimating genus of the zero-divisor graph of a ring. In this note we investigate genera of graphs of a class of zero-divisor rings (a ring in which every element is a zero divisor). We call a ring R to be right absorbing if for a; b in R, ab is not 0, then ab D a. We first show that right absorbing rings are generalized right Klein 4-rings of characteristic two and that these are non-commutative zero-divisor local rings. The zero-divisor graph of such a ring is proved to be precisely the union of a complete graph and a complete bipartite graph. Finally, we have estimated lower and upper bounds of the genus of such a ring.

Graceful Labeling for Some Complete Bipartite Graph

2018 ◽  
Vol 9 (12) ◽  
pp. 2147-2152
Author(s):  
V. Raju ◽  
M. Paruvatha vathana
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graceful Labeling
Sign in / Sign up

Export Citation Format

Share Document