scholarly journals Generalizing the Ramsey Problem through Diameter

10.37236/1658 ◽  
2001 ◽  
Vol 9 (1) ◽  
Author(s):  
Dhruv Mubayi
Keyword(s):  
Prime Power ◽  
Ramsey Problem ◽  
Ramsey Numbers ◽  
The Third ◽  
Positive Integers

Given a graph $G$ and positive integers $d,k$, let $f_d^k(G)$ be the maximum $t$ such that every $k$-coloring of $E(G)$ yields a monochromatic subgraph with diameter at most $d$ on at least $t$ vertices. Determining $f_1^k(K_n)$ is equivalent to determining classical Ramsey numbers for multicolorings. Our results include $\bullet$ determining $f_d^k(K_{a,b})$ within 1 for all $d,k,a,b$ $\bullet$ for $d \ge 4$, $f_d^3(K_n)=\lceil n/2 \rceil +1$ or $\lceil n/2 \rceil$ depending on whether $n \equiv 2 (mod 4)$ or not $\bullet$ $f_3^k(K_n) > {{n}\over {k-1+1/k}}$ The third result is almost sharp, since a construction due to Calkin implies that $f_3^k(K_n) \le {{n}\over {k-1}} +k-1$ when $k-1$ is a prime power. The asymptotics for $f_d^k(K_n)$ remain open when $d=k=3$ and when $d\ge 3, k \ge 4$ are fixed.

scholarly journals Multicolor Ramsey Numbers via Pseudorandom Graphs

10.37236/9071 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Xiaoyu He ◽  
Yuval Wigderson
Keyword(s):  
Spectral Expansion ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Positive Integers

A weakly optimal $K_s$-free $(n,d,\lambda)$-graph is a $d$-regular $K_s$-free graph on $n$ vertices with $d=\Theta(n^{1-\alpha})$ and spectral expansion $\lambda=\Theta(n^{1-(s-1)\alpha})$, for some fixed $\alpha>0$. Such a graph is called optimal if additionally $\alpha = \frac{1}{2s-3}$. We prove that if $s_{1},\ldots,s_{k}\ge3$ are fixed positive integers and weakly optimal $K_{s_{i}}$-free pseudorandom graphs exist for each $1\le i\le k$, then the multicolor Ramsey numbers satisfy\[\Omega\Big(\frac{t^{S+1}}{\log^{2S}t}\Big)\le r(s_{1},\ldots,s_{k},t)\le O\Big(\frac{t^{S+1}}{\log^{S}t}\Big),\]as $t\rightarrow\infty$, where $S=\sum_{i=1}^{k}(s_{i}-2)$. This generalizes previous results of Mubayi and Verstra\"ete, who proved the case $k=1$, and Alon and Rödl, who proved the case $s_1=\cdots = s_k = 3$. Both previous results used the existence of optimal rather than weakly optimal $K_{s_i}$-free graphs.

scholarly journals The third Ramsey numbers for graphs with at most four edges

1994 ◽  
Vol 125 (1-3) ◽  
pp. 399-406 ◽  
Author(s):  
Yuansheng Yang ◽  
Peter Rowlinson
Keyword(s):  
Ramsey Numbers ◽  
The Third

scholarly journals Oscillatory and Asymptotic Behavior of Third Order Half-linear Neutral Dierential Equation with \Maxima"

2017 ◽  
Vol 13 (3) ◽  
pp. 7219-7229
Author(s):  
R Arul ◽  
A Ashok ◽  
G Ayyappan
Keyword(s):  
Asymptotic Behavior ◽  
Third Order ◽  
The Third ◽  
Positive Integers

In this paper we study the oscillation and asymptotic behavior of the third orderneutral dierential equation with \maxima" of the formwhere is the quotient of odd positive integers. Some examples are given toillustrate the main results.

scholarly journals Absolutely split metacyclic groups and weak metacirculants

2019 ◽  
Vol 18 (06) ◽  
pp. 1950117 ◽  
Author(s):  
Li Cui ◽  
Jin-Xin Zhou
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Necessary Condition ◽  
Prime Power Order ◽  
Metacyclic Groups ◽  
Vertex Transitive ◽  
Open Question ◽  
Positive Integers ◽  
The Relationship ◽  
Sufficient And Necessary Condition

Let [Formula: see text] be positive integers, and let [Formula: see text] be a split metacyclic group such that [Formula: see text]. We say that [Formula: see text] is absolutely split with respect to[Formula: see text] provided that for any [Formula: see text], if [Formula: see text], then there exists [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we give a sufficient and necessary condition for the group [Formula: see text] being absolutely split. This generalizes a result of Sanming Zhou and the second author in [Weak metacirculants of odd prime power order, J. Comb. Theory A 155 (2018) 225–243]. We also use this result to investigate the relationship between metacirculants and weak metacirculants. Metacirculants were introduced by Alspach and Parsons in [Formula: see text] and have been a rich source of various topics since then. As a generalization of this class of graphs, Marušič and Šparl in 2008 introduced the so-called weak metacirculants. A graph is called a weak metacirculant if it has a vertex-transitive metacyclic automorphism group. In this paper, it is proved that a weak metacirculant of [Formula: see text]-power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. This provides a partial answer to an open question in the literature.

scholarly journals Landau’s theorem on conjugacy classes for normal subgroups

2016 ◽  
Vol 26 (07) ◽  
pp. 1453-1466
Author(s):  
Antonio Beltrán ◽  
María José Felipe ◽  
Carmen Melchor
Keyword(s):  
Normal Subgroup ◽  
Finite Groups ◽  
Prime Power ◽  
Upper Bounds ◽  
Conjugacy Classes ◽  
Index Formula ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
Small Index ◽  
Positive Integers

Landau’s theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly [Formula: see text] conjugacy classes for any positive integer [Formula: see text]. We show that, for any positive integers [Formula: see text] and [Formula: see text], there exist finitely many finite groups [Formula: see text], up to isomorphism, having a normal subgroup [Formula: see text] of index [Formula: see text] which contains exactly [Formula: see text] non-central [Formula: see text]-conjugacy classes. Upper bounds for the orders of [Formula: see text] and [Formula: see text] are obtained; we use these bounds to classify all finite groups with normal subgroups having a small index and few [Formula: see text]-classes. We also study the related problems when we consider only the set of [Formula: see text]-classes of prime-power order elements contained in a normal subgroup.

scholarly journals A congruence involving harmonic sums modulo pαqβ

2017 ◽  
Vol 13 (05) ◽  
pp. 1083-1094 ◽  
Author(s):  
Tianxin Cai ◽  
Zhongyan Shen ◽  
Lirui Jia
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Prime Factors ◽  
Necessary And Sufficient ◽  
Positive Integers

In 2014, Wang and Cai established the following harmonic congruence for any odd prime [Formula: see text] and positive integer [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] denote the set of positive integers which are prime to [Formula: see text]. In this paper, we obtain an unexpected congruence for distinct odd primes [Formula: see text], [Formula: see text] and positive integers [Formula: see text], [Formula: see text] and the necessary and sufficient condition for [Formula: see text] Finally, we raise a conjecture that for [Formula: see text] and odd prime power [Formula: see text], [Formula: see text], [Formula: see text] However, we fail to prove it even for [Formula: see text] with three distinct prime factors.

scholarly journals Ramsey-type numbers involving graphs and hypergraphs with large girth

2021 ◽  
pp. 1-19
Author(s):  
Hiêp Hàn ◽  
Troy Retter ◽  
Vojtêch Rödl ◽  
Mathias Schacht
Keyword(s):  
Arithmetic Progressions ◽  
Ramsey Number ◽  
Ramsey Property ◽  
Numerical Problem ◽  
The Third ◽  
Ramsey Type ◽  
Large Girth ◽  
Positive Integers

Abstract Erdős asked if, for every pair of positive integers g and k, there exists a graph H having girth (H) = k and the property that every r-colouring of the edges of H yields a monochromatic cycle C k . The existence of such graphs H was confirmed by the third author and Ruciński. We consider the related numerical problem of estimating the order of the smallest graph H with this property for given integers r and k. We show that there exists a graph H on R10k2; k15k3 vertices (where R = R(C k ; r) is the r-colour Ramsey number for the cycle C k ) having girth (H) = k and the Ramsey property that every r-colouring of the edges of H yields a monochromatic C k Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.

scholarly journals Biregular Cages of Girth Five

10.37236/2594 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Marién Abreu ◽  
Gabriela Araujo-Pardo ◽  
Camino Balbuena ◽  
Domenico Labbate ◽  
Gloria López-Chávez
Keyword(s):  
Prime Power ◽  
Discrete Math ◽  
Regular Graphs ◽  
Minimum Order ◽  
Degree Set ◽  
Positive Integers

Let $2 \le r < m$ and $g$ be positive integers. An $(\{r,m\};g)$-graph (or biregular graph) is a graph with degree set $\{r,m\}$ and girth $g$, and an $(\{r,m\};g)$-cage (or biregular cage) is an $(\{r,m\};g)$-graph of minimum order $n(\{r,m\};g)$. If $m=r+1$, an $(\{r,m\};g)$-cage is said to be a semiregular cage.In this paper we generalize the reduction and graph amalgam operations from [M. Abreu,  G. Araujo-Pardo, C. Balbuena, D. Labbate. Families of Small Regular Graphs of Girth $5$. Discrete Math. 312 (2012) 2832--2842] on the incidence graphs of an affine and a biaffine plane obtaining two new infinite families of biregular cages and two new semiregular cages. The constructed new families are $(\{r,2r-3\};5)$-cages for all $r=q+1$ with $q$ a prime power, and $(\{r,2r-5\};5)$-cages for all $r=q+1$ with $q$ a prime. The new semiregular cages are constructed for $r=5$ and $6$ with $31$ and $43$ vertices respectively.

scholarly journals Symbol-Pair Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths over \({{\mathbb{F}_{p^m}[u]}/{\langle u^3\rangle}}\)

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2554
Author(s):  
Mohammed E. Charkani ◽  
Hai Q. Dinh ◽  
Jamal Laaouine ◽  
Woraphon Yamaka
Keyword(s):  
Finite Field ◽  
Prime Power ◽  
Nonzero Element ◽  
Constacyclic Codes ◽  
Chain Ring ◽  
Positive Integers

Let p be a prime, s, m be positive integers, γ be a nonzero element of the finite field Fpm, and let R=Fpm[u]/⟨u3⟩ be the finite commutative chain ring. In this paper, the symbol-pair distances of all γ-constacyclic codes of length ps over R are completely determined.

Oscillation of certain third-order quasilinear neutral differential equations

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Linlin Yang ◽  
Zhiting Xu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Neutral Differential Equation ◽  
Third Order ◽  
Oscillation Criteria ◽  
Neutral Differential Equations ◽  
The Third ◽  
Positive Integers ◽  
Type Oscillation

AbstractIn this paper, new oscillation criteria for the third-order quasilinear neutral differential equation $$\left( {a\left( t \right)\left( {z''\left( t \right)} \right)^\gamma } \right)^\prime + q\left( t \right)x^\gamma \left( {\tau \left( t \right)} \right) = 0, t \geqslant t_0 ,$$ are established, where z(t) = x(t) + p(t)x(δ(t)), and γ is a ratio of odd positive integers. Those results extend the oscillation criteria due to Sun [SUN, Y. G.: New Kamenev-type oscillation criteria for second-order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 (2004) 341–351] to the equation, and complement the existing results in literature. Two examples are provided to illustrate the relevance of our main theorems.

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