scholarly journals Generalized Ramsey Numbers: Forbidding Paths with Few Colors

10.37236/8801 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Robert A. Krueger
Keyword(s):  
Asymptotic Behavior ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Extremal Function ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
Almost All

Let $f(K_n, H, q)$ be the minimum number of colors needed to edge-color $K_n$ so that every copy of $H$ is colored with at least $q$ colors. Originally posed by Erdős and Shelah when $H$ is complete, the asymptotics of this extremal function have been extensively studied when $H$ is a complete graph or a complete balanced bipartite graph. Here we investigate this function for some other $H$, and in particular we determine the asymptotic behavior of $f(K_n, P_v, q)$ for almost all values of $v$ and $q$, where $P_v$ is a path on $v$ vertices.

scholarly journals Asymptotic bounds for some bipartite graph: complete graph Ramsey numbers

2000 ◽  
Vol 220 (1-3) ◽  
pp. 51-56 ◽  
Author(s):  
Yair Caro ◽  
Yusheng Li ◽  
Cecil C. Rousseau ◽  
Yuming Zhang
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Ramsey Numbers ◽  
Asymptotic Bounds

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 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 Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

scholarly journals A Note on On-Line Ramsey Numbers for Some Paths

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 735
Author(s):  
Tomasz Dzido ◽  
Renata Zakrzewska
Keyword(s):  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
On Line ◽  
Infinite Set ◽  
Do So

We consider the important generalisation of Ramsey numbers, namely on-line Ramsey numbers. It is easiest to understand them by considering a game between two players, a Builder and Painter, on an infinite set of vertices. In each round, the Builder joins two non-adjacent vertices with an edge, and the Painter colors the edge red or blue. An on-line Ramsey number r˜(G,H) is the minimum number of rounds it takes the Builder to force the Painter to create a red copy of graph G or a blue copy of graph H, assuming that both the Builder and Painter play perfectly. The Painter’s goal is to resist to do so for as long as possible. In this paper, we consider the case where G is a path P4 and H is a path P10 or P11.

scholarly journals Turán theorems for unavoidable patterns

2021 ◽  
pp. 1-20
Author(s):  
ANTÓNIO GIRÃO ◽  
BHARGAV NARAYANAN
Keyword(s):  
Complete Graph ◽  
Blow Up ◽  
Ramsey Numbers ◽  
Order Of Magnitude ◽  
Transitive Tournament ◽  
Unavoidable Patterns

Abstract We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for t ≥ 3, we show that any two-colouring of the complete graph on n vertices that is δ-far from being monochromatic contains an unavoidable t-colouring when δ ≫ n−1/t, where an unavoidable t-colouring is any two-colouring of a clique of order 2t in which one colour forms either a clique of order t or two disjoint cliques of order t. Next, for t ≥ 3, we show that any tournament on n vertices that is δ-far from being transitive contains an unavoidable t-tournament when δ ≫ n−1/[t/2], where an unavoidable t-tournament is the blow-up of a cyclic triangle obtained by replacing each vertex of the triangle by a transitive tournament of order t. Conditional on a well-known conjecture about bipartite Turán numbers, both our results are sharp up to implied constants and hence determine the order of magnitude of the corresponding off-diagonal Ramsey numbers.

scholarly journals Ramsey Numbers for Theta Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
M. M. M. Jaradat ◽  
M. S. A. Bataineh ◽  
S. M. E. Radaideh
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Subgraph Isomorphic

The graph Ramsey number is the smallest integer with the property that any complete graph of at least vertices whose edges are colored with two colors (say, red and blue) contains either a subgraph isomorphic to all of whose edges are red or a subgraph isomorphic to all of whose edges are blue. In this paper, we consider the Ramsey numbers for theta graphs. We determine , for . More specifically, we establish that for . Furthermore, we determine for . In fact, we establish that if is even, if is odd.

scholarly journals An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 525
Author(s):  
Javier Rodrigo ◽  
Susana Merchán ◽  
Danilo Magistrali ◽  
Mariló López
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
General Position ◽  
Crossing Number ◽  
Minimum Number

In this paper, we improve the lower bound on the minimum number of  ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties

1969 ◽  
Vol 21 ◽  
pp. 992-1000 ◽  
Author(s):  
L. W. Beineke
Keyword(s):  
Projective Plane ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Graphs ◽  
Minimum Number ◽  
Double Torus

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

scholarly journals Sidorenko's Conjecture, Colorings and Independent Sets

10.37236/6019 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Péter Csikvári ◽  
Zhicong Lin
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Independent Sets ◽  
Correlation Inequality

Let $\hom(H,G)$ denote the number of homomorphisms from a graph $H$ to a graph $G$. Sidorenko's conjecture asserts that for any bipartite graph $H$, and a graph $G$ we have$$\hom(H,G)\geq v(G)^{v(H)}\left(\frac{\hom(K_2,G)}{v(G)^2}\right)^{e(H)},$$where $v(H),v(G)$ and $e(H),e(G)$ denote the number of vertices and edges of the graph $H$ and $G$, respectively. In this paper we prove Sidorenko's conjecture for certain special graphs $G$: for the complete graph $K_q$ on $q$ vertices, for a $K_2$ with a loop added at one of the end vertices, and for a path on $3$ vertices with a loop added at each vertex. These cases correspond to counting colorings, independent sets and Widom-Rowlinson colorings of a graph $H$. For instance, for a bipartite graph $H$ the number of $q$-colorings $\mathrm{ch}(H,q)$ satisfies$$\mathrm{ch}(H,q)\geq q^{v(H)}\left(\frac{q-1}{q}\right)^{e(H)}.$$In fact, we will prove that in the last two cases (independent sets and Widom-Rowlinson colorings) the graph $H$ does not need to be bipartite. In all cases, we first prove a certain correlation inequality which implies Sidorenko's conjecture in a stronger form.

Sign in / Sign up

Export Citation Format

Share Document