scholarly journals Off-Diagonal Ordered Ramsey Numbers of Matchings

10.37236/8085 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Dhruv Rohatgi
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Edge Coloring ◽  
Ramsey Number ◽  
Relative Order ◽  
Ramsey Numbers ◽  
Asymptotic Bounds ◽  
Special Cases ◽  
Ordered Graphs

For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete ordered graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the off-diagonal ordered Ramsey number $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $\max r_<(M, K_3)$, where the maximum is over all ordered matchings $M$ on $n$ vertices. The best-known upper bound is $O(n^2/\log n)$, whereas the best-known lower bound is $\Omega((n/\log n)^{4/3})$, and Conlon et al. hypothesize that there is some fixed $\epsilon > 0$ such that $r_<(M, K_3) = O(n^{2-\epsilon})$ for every ordered matching $M$. We resolve two special cases of this conjecture. We show that the off-diagonal ordered Ramsey numbers for ordered matchings in which edges do not cross are nearly linear. We also prove a truly sub-quadratic upper bound for random ordered matchings with interval chromatic number $2$.

scholarly journals Induced Ramsey Number for a Star Versus a Fixed Graph

10.37236/9358 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Maria Axenovich ◽  
Izolda Gorgol
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Constant Factor ◽  
Ramsey Number ◽  
Large N ◽  
Star Versus

We write $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$ for graphs $F, G,$ and $H$, if for any coloring of the edges of $F$ in red and blue, there is either a red induced copy of $H$ or a blue induced copy of $G$. For graphs $G$ and $H$, let $\mathrm{IR}(H,G)$ be the smallest number of vertices in a graph $F$ such that $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$. In this note we consider the case when $G$ is a star on $n$ edges, for large $n$ and $H$ is a fixed graph. We prove that  $$ (\chi(H)-1) n \leq \mathrm{IR}(H, K_{1,n}) \leq (\chi(H)-1)^2n + \epsilon n,$$ for any $\epsilon>0$,  sufficiently large $n$, and $\chi(H)$ denoting the chromatic number of $H$. The lower bound is asymptotically tight  for any fixed bipartite $H$. The upper bound is attained up to a constant factor, for example when $H$ is a clique.

scholarly journals Ramsey Numbers of Ordered Graphs

10.37236/7816 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Martin Balko ◽  
Josef Cibulka ◽  
Karel Král ◽  
Jan Kynčl
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Positive Answer ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
Ordered Graphs ◽  
Constant Interval ◽  
Monochromatic Copy ◽  
Special Classes

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.

scholarly journals Batas Bawah Bilangan Ramsey pada Graf Bintang S_10 Versus Roda W_12 Lower Bound of the Ramsey Number for the Star Graph S_10 Versus Wheels W_12

2018 ◽  
Vol 3 (1) ◽  
pp. 471
Author(s):  
Hamdana Hadaming ◽  
Andi Ardhila Wahyudi
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Lower Boundary ◽  
Ramsey Number ◽  
Star Graph ◽  
Point Graph ◽  
Ramsey Numbers ◽  
Critical Graph ◽  
Star Versus

Bilangan Ramsey untuk graf  terhadap graf , dinotasikan dengan  adalah bilangan bulat terkecil  sedemikian sehingga untuk setiap graf  dengan orde akan memenuhi sifat berikut:  memuat graf  atau komplemen dari  memuat graf .Penelitian ini bertujuan untuk  menentukan graf kritis maksimum  dan  dengan genap. Berdasarkan batas bawah tersebut di tentukan batas atas minimum sehingga diperoleh nilai bilangan Ramsey untuk graf bintang  versus , atau . Dengan demikian penentuan batas bawah bilangan Ramsey  dilakukan dengan cara batas bawah yang  diberikan oleh Chavatal dan Harary, untuk bilangan Ramsey pada graf bintang  versus  adalah , dengan  adalah bilangan kromatik titik graf roda  dan  adalah kardinalitas komponen terbesar graf . Berdasarkan batas bawah Chavatal dan Harary tersebut dikonstruksi graf kritis untuk  dan  yang ordenya lebih besar dari nilai batas bawah yang diberikan Chavatal dan Harary. Orde dari graf kritis tersebut merupakan batas bawah terbaik untuk . Kata kunci: Bilangan Ramsey, bintang, roda AbstractRamsey Numbers for a graph  to a graph , denoted by   is the smallest integer n such that for every graph  of order  either  the following meeet:  contains a graph  or the complement of  contains the graph . This aims of the study to determine the maximum critical graph  and . Based on the lower bound of the specified minimum upper bound in order to obtain numerical values for the Ramsey graph  Star versus  , or . Thus the determination of Ramsey numbers .  is done by determine the lower boundary and upper bound. The lower bound given by Chavatal and Harary, for ramsey number for star graph versus wheel   is , is a point graph of chromatic number wheel  and  is the cardinality of the largest component of the graph . Based on the lower bound Chavatal and Harary graph is constructed critical to  and  are poin greater than the lower bound value given Chavatal and Harary. Order of the critical graph is the best lower bound for . Keywords : Ramsey number, Stars, and Wheels

scholarly journals A Note on Restricted Online Ramsey Numbers of Matchings

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Vojtěch Dvořák
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Short Note ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number

Consider the following game between Builder and Painter. We take some families of graphs $\mathcal{G}_{1},\ldots,\mathcal{G}_t$ and an integer $n$ such that $n \geq R(\mathcal{G}_1,\ldots,\mathcal{G}_t)$. In each turn, Builder picks an edge of initially uncoloured $K_n$ and Painter colours that edge with some colour $i \in \left\{ 1,\ldots,t \right\}$ of her choice. The game ends when a graph $G_i$ in colour $i $ for some $G_i \in \mathcal{G}_i$ and some $i$ is created. The restricted online Ramsey number $\tilde{R}(\mathcal{G}_{1},\ldots,\mathcal{G}_t;n)$ is the minimum number of turns that Builder needs to guarantee the game to end. In a recent paper, Briggs and Cox studied the restricted online Ramsey numbers of matchings and determined a general upper bound for them. They proved that for $n=3r-1=R_2(r K_2)$ we have $\tilde{R}_{2}(r K_2;n) \leq n-1$ and asked whether this was tight. In this short note, we provide a general lower bound for these Ramsey numbers. As a corollary, we answer this question of Briggs and Cox, and confirm that for $n=3r-1$ we have $\tilde{R}_{2}(r K_2;n) = n-1$. We also show that for $n'=4r-2=R_3(r K_2)$ we have $\tilde{R}_{3}(r K_2;n') = 5r-4$.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

scholarly journals On r-dynamic coloring of comb graphs

2021 ◽  
Vol 27 (2) ◽  
pp. 191-200
Author(s):  
K. Kalaiselvi ◽  
◽  
N. Mohanapriya ◽  
J. Vernold Vivin ◽  
◽  
...  
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Middle Graph ◽  
Different Color

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

scholarly journals Minimum Clique Number, Chromatic Number, and Ramsey Numbers

10.37236/2125 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Gaku Liu
Keyword(s):  
Chromatic Number ◽  
Asymptotic Properties ◽  
Inverse Function ◽  
Clique Number ◽  
Ramsey Number ◽  
Growth Order ◽  
Ramsey Numbers

Let $Q(n,c)$ denote the minimum clique number over graphs with $n$ vertices and chromatic number $c$. We investigate the asymptotics of $Q(n,c)$ when $n/c$ is held constant. We show that when $n/c$ is an integer $\alpha$, $Q(n,c)$ has the same growth order as the inverse function of the Ramsey number $R(\alpha+1,t)$ (as a function of $t$). Furthermore, we show that if certain asymptotic properties of the Ramsey numbers hold, then $Q(n,c)$ is in fact asymptotically equivalent to the aforementioned inverse function. We use this fact to deduce that $Q(n,\lceil n/3 \rceil)$ is asymptotically equivalent to the inverse function of $R(4,t)$.

scholarly journals Constrained Ramsey Numbers

2009 ◽  
Vol 18 (1-2) ◽  
pp. 247-258 ◽  
Author(s):  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Ramsey Theorem ◽  
Logarithmic Factor ◽  
Subgraph Isomorphic ◽  
Rate Of Growth ◽  
Special Cases ◽  
Edge Colouring

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge colouring of the complete graph on n vertices (with any number of colours) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T. Here, a subgraph is said to be rainbow if all of its edges have different colours. It is an immediate consequence of the Erdős–Rado Canonical Ramsey Theorem that f(S, T) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang and Ling showed that f(S, T) ≤ O(st2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this paper, we study this case and show that f(S, Pt) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.

scholarly journals A Note on Edge-Colourings Avoiding Rainbow $K_4$ and Monochromatic $K_m$

10.37236/257 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Veselin Jungić ◽  
Tomáš Kaiser ◽  
Daniel Král'
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Projective Planes ◽  
Ramsey Number ◽  
Complete Subgraph ◽  
Finite Projective Planes ◽  
Edge Colourings ◽  
Edge Colouring

We study the mixed Ramsey number $maxR(n,{K_m},{K_r})$, defined as the maximum number of colours in an edge-colouring of the complete graph $K_n$, such that $K_n$ has no monochromatic complete subgraph on $m$ vertices and no rainbow complete subgraph on $r$ vertices. Improving an upper bound of Axenovich and Iverson, we show that $maxR(n,{K_m},{K_4}) \leq n^{3/2}\sqrt{2m}$ for all $m\geq 3$. Further, we discuss a possible way to improve their lower bound on $maxR(n,{K_4},{K_4})$ based on incidence graphs of finite projective planes.

scholarly journals A Note on Odd Cycle-Complete Graph Ramsey Numbers

10.37236/1662 ◽  
2001 ◽  
Vol 9 (1) ◽  
Author(s):  
Benny Sudakov
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Upper Bound ◽  
Short Note ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Odd Cycle

The Ramsey number $r(C_l, K_n)$ is the smallest positive integer $m$ such that every graph of order $m$ contains either cycle of length $l$ or a set of $n$ independent vertices. In this short note we slightly improve the best known upper bound on $r(C_l, K_n)$ for odd $l$.

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