scholarly journals Trees with an On-Line Degree Ramsey Number of Four

10.37236/660 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
David Rolnick
Keyword(s):  
Maximum Degree ◽  
Ramsey Theory ◽  
Complete Classification ◽  
Ramsey Number ◽  
On Line ◽  
Delta G ◽  
Goal Graph ◽  
Monochromatic Copy

On-line Ramsey theory studies a graph-building game between two players. The player called Builder builds edges one at a time, and the player called Painter paints each new edge red or blue after it is built. The graph constructed is called the background graph. Builder's goal is to cause the background graph to contain a monochromatic copy of a given goal graph, and Painter's goal is to prevent this. In the $S_k$-game variant of the typical game, the background graph is constrained to have maximum degree no greater than $k$. The on-line degree Ramsey number $\mathring{R}_{\Delta}(G)$ of a graph $G$ is the minimum $k$ such that Builder wins an $S_k$-game in which $G$ is the goal graph. Butterfield et al. previously determined all graphs $G$ satisfying $\mathring{R}_{\Delta}(G)\le 3$. We provide a complete classification of trees $T$ satisfying $\mathring{R}_{\Delta}(T)=4$.

scholarly journals On-line Ramsey Theory for Bounded Degree Graphs

10.37236/623 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jane Butterfield ◽  
Tracy Grauman ◽  
William B. Kinnersley ◽  
Kevin G. Milans ◽  
Christopher Stocker ◽  
...  
Keyword(s):  
Ramsey Theory ◽  
Upper Bounds ◽  
Ramsey Number ◽  
Bounded Degree ◽  
Linear Forest ◽  
On Line ◽  
Bounded Degree Graphs ◽  
Vertex Degrees ◽  
Delta G ◽  
Monochromatic Copy

When graph Ramsey theory is viewed as a game, "Painter" 2-colors the edges of a graph presented by "Builder". Builder wins if every coloring has a monochromatic copy of a fixed graph $G$. In the on-line version, iteratively, Builder presents one edge and Painter must color it. Builder must keep the presented graph in a class ${\cal H}$. Builder wins the game $(G,{\cal H})$ if a monochromatic copy of $G$ can be forced. The on-line degree Ramsey number $\mathring {R}_\Delta(G)$ is the least $k$ such that Builder wins $(G,{\cal H})$ when ${\mathcal H}$ is the class of graphs with maximum degree at most $k$. Our results include: 1) $\mathring {R}_\Delta(G)\!\le\!3$ if and only if $G$ is a linear forest or each component lies inside $K_{1,3}$. 2) $\mathring {R}_\Delta(G)\ge \Delta(G)+t-1$, where $t=\max_{uv\in E(G)}\min\{d(u),d(v)\}$. 3) $\mathring {R}_\Delta(G)\le d_1+d_2-1$ for a tree $G$, where $d_1$ and $d_2$ are two largest vertex degrees. 4) $4\le \mathring {R}_\Delta(C_n)\le 5$, with $\mathring {R}_\Delta(C_n)=4$ except for finitely many odd values of $n$. 5) $\mathring {R}_\Delta(G)\le6$ when $\Delta(G)\le 2$. The lower bounds come from strategies for Painter that color edges red whenever the red graph remains in a specified class. The upper bounds use a result showing that Builder may assume that Painter plays "consistently".

scholarly journals On-line Ramsey Theory

10.37236/1810 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
J. A. Grytczuk ◽  
M. Hałuszczak ◽  
H. A. Kierstead
Keyword(s):  
Planar Graphs ◽  
Ramsey Theory ◽  
Winning Strategy ◽  
Fixed Target ◽  
Outerplanar Graphs ◽  
On Line ◽  
Ramsey Type ◽  
Colorable Graph ◽  
Monochromatic Copy

The Ramsey game we consider in this paper is played on an unbounded set of vertices by two players, called Builder and Painter. In one move Builder introduces a new edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed target graph $H$, keeping the constructed graph in a prescribed class ${\cal G}$. The main problem is to recognize the winner for a given pair $H,{\cal G}$. In particular, we prove that Builder has a winning strategy for any $k$-colorable graph $H$ in the game played on $k$-colorable graphs. Another class of graphs with this strange self-unavoidability property is the class of forests. We show that the class of outerplanar graphs does not have this property. The question of whether planar graphs are self-unavoidable is left open. We also consider a multicolor version of Ramsey on-line game. To extend our main result for $3$-colorable graphs we introduce another Ramsey type game, which seems interesting in its own right.

scholarly journals Degree Ramsey Numbers of Closed Blowups of Trees

10.37236/2526 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Paul Horn ◽  
Kevin G. Milans ◽  
Vojtěch Rödl
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Edge Coloring ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Delta G ◽  
Monochromatic Copy

The degree Ramsey number of a graph $G$, denoted $R_\Delta(G;s)$, is $\min\{\Delta(H)\colon\, H\stackrel{s}{\to} G\}$, where $H\stackrel{s}{\to} G$ means that every $s$-edge-coloring of $H$ contains a monochromatic copy of $G$.  The closed $k$-blowup of a graph is obtained by replacing every vertex with a clique of size $k$ and every edge with a complete bipartite graph where both partite sets have size $k$.  We prove that there is a function $f$ such that $R_\Delta(G;s) \le f(\Delta(G), s)$  when $G$ is a closed blowup of a tree.

scholarly journals Degree Ramsey Numbers of Graphs

2012 ◽  
Vol 21 (1-2) ◽  
pp. 229-253 ◽  
Author(s):  
WILLIAM B. KINNERSLEY ◽  
KEVIN G. MILANS ◽  
DOUGLAS B. WEST
Keyword(s):  
Maximum Degree ◽  
Double Star ◽  
Ramsey Number ◽  
Bounded Degree ◽  
Ramsey Numbers ◽  
Colour Class ◽  
Bounded Degree Graphs ◽  
Edge Colourings ◽  
Edge Colouring ◽  
Monochromatic Copy

Let HG mean that every s-colouring of E(H) produces a monochromatic copy of G in some colour class. Let the s-colour degree Ramsey number of a graph G, written RΔ(G; s), be min{Δ(H): HG}. If T is a tree in which one vertex has degree at most k and all others have degree at most ⌈k/2⌉, then RΔ(T; s) = s(k − 1) + ϵ, where ϵ = 1 when k is odd and ϵ = 0 when k is even. For general trees, RΔ(T; s) ≤ 2s(Δ(T) − 1).To study sharpness of the upper bound, consider the double-starSa,b, the tree whose two non-leaf vertices have degrees a and b. If a ≤ b, then RΔ(Sa,b; 2) is 2b − 2 when a < b and b is even; it is 2b − 1 otherwise. If s is fixed and at least 3, then RΔ(Sb,b;s) = f(s)(b − 1) − o(b), where f(s) = 2s − 3.5 − O(s−1).We prove several results about edge-colourings of bounded-degree graphs that are related to degree Ramsey numbers of paths. Finally, for cycles we show that RΔ(C2k + 1; s) ≥ 2s + 1, that RΔ(C2k; s) ≥ 2s, and that RΔ(C4;2) = 5. For the latter we prove the stronger statement that every graph with maximum degree at most 4 has a 2-edge-colouring such that the subgraph in each colour class has girth at least 5.

scholarly journals On-line Ramsey numbers for paths and stars

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
J. A. Grytczuk ◽  
H. A. Kierstead ◽  
P. Prałat
Keyword(s):  
Upper Bound ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
International Audience ◽  
On Line ◽  
Monochromatic Copy

Graphs and Algorithms International audience We study on-line version of size-Ramsey numbers of graphs defined via a game played between Builder and Painter: in one round Builder joins two vertices by an edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number r(H) of the graph H. We determine exact values of r(H) for a few short paths and obtain a general upper bound r(Pn) ≤ 4n −7. We also study asymmetric version of this parameter when one of the target graphs is a star Sn with n edges. We prove that r(Sn, H) ≤ n*e(H) when H is any tree, cycle or clique

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 Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

scholarly journals Characteristic cohomology and observables in higher spin gravity

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Alexey Sharapov ◽  
Evgeny Skvortsov
Keyword(s):  
Higher Spin Gravity ◽  
Complete Classification ◽  
Gravity Models ◽  
Simons Theory ◽  
Surface Charges ◽  
Higher Spin ◽  
Dynamical Invariants ◽  
Chern Simons ◽  
Conserved Currents

Abstract We give a complete classification of dynamical invariants in 3d and 4d Higher Spin Gravity models, with some comments on arbitrary d. These include holographic correlation functions, interaction vertices, on-shell actions, conserved currents, surface charges, and some others. Surprisingly, there are a good many conserved p-form currents with various p. The last fact, being in tension with ‘no nontrivial conserved currents in quantum gravity’ and similar statements, gives an indication of hidden integrability of the models. Our results rely on a systematic computation of Hochschild, cyclic, and Chevalley-Eilenberg cohomology for the corresponding higher spin algebras. A new invariant in Chern-Simons theory with the Weyl algebra as gauge algebra is also presented.

scholarly journals Nil-clean quadratic elements

2017 ◽  
Vol 16 (10) ◽  
pp. 1750197 ◽  
Author(s):  
Janez Šter
Keyword(s):  
Complete Classification ◽  
Strong Condition ◽  
Clean Rings

We provide a strong condition holding for nil-clean quadratic elements in any ring. In particular, our result implies that every nil-clean involution in a ring is unipotent. As a consequence, we give a complete classification of weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl. 15 (2016) 1650148, doi: 10.1142/S0219498816501486].

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