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 $W_{2k+1}$.

scholarly journals A Turán Type Problem Concerning the Powers of the Degrees of a Graph

10.37236/1525 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Degree Sequence ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Type Problem ◽  
Maximum Value ◽  
Turán Number

For a graph $G$ whose degree sequence is $d_{1},\ldots ,d_{n}$, and for a positive integer $p$, let $e_{p}(G)=\sum_{i=1}^{n}d_{i}^{p}$. For a fixed graph $H$, let $t_{p}(n,H)$ denote the maximum value of $e_{p}(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $t_{1}(n,H)$ is twice the Turán number of $H$. In this paper we consider the case $p>1$. For some graphs $H$ we obtain exact results, for some others we can obtain asymptotically tight upper and lower bounds, and many interesting cases remain open.

scholarly journals Further results on a generalization of Bertrand's postulate

1996 ◽  
Vol 19 (1) ◽  
pp. 75-85
Author(s):  
George Giordano
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Bertrand’S Postulate

Letd(k)be defined as the least positive integernfor whichpn+1<2pn−k. In this paper we will show that fork≥286664, thend(k)<k/(logk−2.531)and fork≥2, thenk(1−1/logk)/logk<d(k). Furthermore, forksufficiently large we establish upper and lower bounds ford(k).

A Note on the Multi-Colored Bipartite Ramsey Number for Paths

2021 ◽  
pp. 2150041
Author(s):  
Hanxiao Qiao ◽  
Ke Wang ◽  
Suonan Renqian ◽  
Renqingcuo
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Bipartite Graphs ◽  
Ramsey Number ◽  
Upper And Lower Bounds ◽  
Number Formula

For bipartite graphs [Formula: see text], the bipartite Ramsey number [Formula: see text] is the least positive integer [Formula: see text] so that any coloring of the edges of [Formula: see text] with [Formula: see text] colors will result in a copy of [Formula: see text] in the [Formula: see text]th color for some [Formula: see text]. In this paper, we get the exact value of [Formula: see text], and obtain the upper and lower bounds of [Formula: see text], where [Formula: see text] denotes a path with [Formula: see text] vertices.

scholarly journals Monochromatic arithmetic progressions with large differences

1999 ◽  
Vol 60 (1) ◽  
pp. 21-35
Author(s):  
Tom C. Brown ◽  
Bruce M. Landman
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Arithmetic Progression ◽  
Constant Function ◽  
Arithmetic Progressions ◽  
Upper And Lower Bounds

A generalisation of the van der Waerden numbers w(k, r) is considered. For a function f: Z+ → R+ define w(f, k, r) to be the least positive integer (if it exists) such that for every r-coloring of [1, w(f, k, r)] there is a monochromatic arithmetic progression {a + id: 0 ≤ i ≤ k −1} such that d ≥ f(a). Upper and lower bounds are given for w(f, 3, 2). For k > 3 or r > 2, particular functions f are given such that w(f, k, r) does not exist. More results are obtained for the case in which f is a constant function.

The Distribution of Zeros of Quadratic Forms over Finite Fields

2015 ◽  
Vol 7 (2) ◽  
pp. 18
Author(s):  
Ali H. Hakami
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Quadratic Forms ◽  
Distribution Of Zeros ◽  
Linearly Independent ◽  
Set Of Points

Let $m$ be a positive integer with $m &lt; p/2$ and $p$ is a prime. Let $\mathbb{F}_q$ be the finite field in $q = p^f$ elements, $Q({\mathbf{x}})$ be a nonsinqular quadratic form over $\mathbb{F}_q$ with $q$ odd, $V$ be the set of points in $\mathbb{F}_q^n$ satisfying the equation $Q({\mathbf{x}}) = 0$ in which the variables are restricted to a box of points of the type\[\mathcal{B}(m) = \left\{ {{\mathbf{x}} \in \mathbb{F}_q^n \left| {x_i  = \sum\limits_{j = 1}^f {x_{ij} \xi _j } ,\;\left| {x_{ij} } \right| &lt; m,\;1 \leqslant i \leqslant n,\;1 \leqslant j \leqslant f} \right.} \right\},\]where $\xi _1 , \ldots ,\xi _f$ is a basis for $\mathbb{F}_q$ over $\mathbb{F}_p$ and $n &gt; 2$ even. Set $\Delta  = \det Q$ such that $\chi \left( {( - 1)^{n/2} \Delta } \right) = 1.$ We shall motivate work of (Cochrane, 1986) to obtain lower bounds on $m,$ size of the box $\mathcal{B},$ so that $\mathcal{B} \cap V$ is nonempty. For this we show that the box $\mathcal{B}(m)$ contains a zero of $Q({\mathbf{x}})$ provided that $m \geqslant p^{1/2}.$ We also show that the box $\mathcal{B}(m)$ contains $n$ linearly independent zeros of $Q({\mathbf{x}})$ provided that $m \geqslant 2^{n/2} p^{1/2} .$

Upper and Lower Bounds for a Function Related to Brown's Lemma

Integers ◽  
2010 ◽  
Vol 10 (6) ◽  
Author(s):  
Hayri Ardal
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Fixed Positive Integer ◽  
Positive Integers

AbstractThe well-known Brown's lemma says that for every finite coloring of the positive integers, there exist a fixed positive integer

Constructions of rotated lattice constellations in dimensions power of 3

2018 ◽  
Vol 17 (09) ◽  
pp. 1850175 ◽  
Author(s):  
Agnaldo José Ferrari ◽  
Antonio Aparecido de Andrade
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Fading Channels ◽  
Lower Bounds ◽  
Rayleigh Fading ◽  
Signal Transmission ◽  
Number Fields ◽  
Rayleigh Fading Channels ◽  
Full Diversity ◽  
Totally Real

In this paper, we present the constructions of rotated [Formula: see text]-lattices, where [Formula: see text] is a positive integer, via [Formula: see text]-modules of the ring of the integers [Formula: see text]. Our focus is on totally real number fields since the associated lattices have full diversity and then may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. Lower bounds for the minimum product distances of such construction are also presented.

scholarly journals Major n-connected graphs

1989 ◽  
Vol 47 (1) ◽  
pp. 43-52
Author(s):  
Ortrud R. Oellermann
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Connected Subgraph ◽  
Edge Connectivity ◽  
Maximum Order ◽  
Induced Subgraph ◽  
Connected Graphs

AbstractAn induced subgraph H of connectivity (edge-connectivity) n in a graph G is a major n-connected (major n-edge-connected) subgraph of G if H contains no subgraph with connectivity (edge- connectivity) exceeding n and H has maximum order with respect to this property. An induced subgraph is a major (major edge-) subgraph if it is a major n-connected (major n-edge-connected) subgraph for some n. Let m be the maximum order among all major subgraphs of C. Then the major connectivity set K(G) of G is defined as the set of all n for which there exists a major n-connected subgraph of G having order m. The major edge-connectivity set is defined analogously. The connectivity and the elements of the major connectivity set of a graph are compared, as are the elements of the major connectivity set and the major edge-connectivity set of a graph. It is shown that every set S of nonnegative integers is the major connectivity set of some graph G. Further, it is shown that for each positive integer m exceeding every element of S, there exists a graph G such that every major k-connected subgraph of G, where k ∈ K(G), has order m. Moreover, upper and lower bounds on the order of such graphs G are established.

scholarly journals COUNTING THE NUMBER OF SOLUTIONS TO THE ERDŐS–STRAUS EQUATION ON UNIT FRACTIONS

2013 ◽  
Vol 94 (1) ◽  
pp. 50-105 ◽  
Author(s):  
CHRISTIAN ELSHOLTZ ◽  
TERENCE TAO
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Diophantine Equation ◽  
Upper And Lower Bounds ◽  
Natural Numbers ◽  
Number Of Solutions ◽  
Waring’S Problem ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

AbstractFor any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$ with $x, y, z$ positive integers. The Erdős–Straus conjecture asserts that $f(n)\gt 0$ for every $n\geq 2$. In this paper we obtain a number of upper and lower bounds for $f(n)$ or $f(p)$ for typical values of natural numbers $n$ and primes $p$. For instance, we establish that $$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$ These upper and lower bounds show that a typical prime has a small number of solutions to the Erdős–Straus Diophantine equation; small, when compared with other additive problems, like Waring’s problem.

scholarly journals Spanning Trees with Many Leaves and Average Distance

10.37236/757 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ermelinda DeLaViña ◽  
Bill Waller
Keyword(s):  
Lower Bounds ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Average Distance ◽  
Independence Number ◽  
Local Independence ◽  
Open Problems ◽  
Maximum Order ◽  
Bipartite Subgraph ◽  
Number Of Leaves

In this paper we prove several new lower bounds on the maximum number of leaves of a spanning tree of a graph related to its order, independence number, local independence number, and the maximum order of a bipartite subgraph. These new lower bounds were conjectured by the program Graffiti.pc, a variant of the program Graffiti. We use two of these results to give two partial resolutions of conjecture 747 of Graffiti (circa 1992), which states that the average distance of a graph is not more than half the maximum order of an induced bipartite subgraph. If correct, this conjecture would generalize conjecture number 2 of Graffiti, which states that the average distance is not more than the independence number. Conjecture number 2 was first proved by F. Chung. In particular, we show that the average distance is less than half the maximum order of a bipartite subgraph, plus one-half; we also show that if the local independence number is at least five, then the average distance is less than half the maximum order of a bipartite subgraph. In conclusion, we give some open problems related to average distance or the maximum number of leaves of a spanning tree.

Sign in / Sign up

Export Citation Format

Share Document