scholarly journals The Game Chromatic Number of Dense Random Graphs

10.37236/4391 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Ralph Keusch ◽  
Angelika Steger
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
High Probability ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Game Chromatic Number ◽  
Order Of Magnitude ◽  
Random Structures

Suppose that two players take turns coloring the vertices of a given graph G with k colors. In each move the current player colors a vertex such that neighboring vertices get different colors. The first player wins this game if and only if at the end, all vertices are colored. The game chromatic number χg(G) is defined as the smallest k for which the first player has a winning strategy.Recently, Bohman, Frieze and Sudakov [Random Structures and Algorithms 2008] analysed the game chromatic number of random graphs and obtained lower and upper bounds of the same order of magnitude. In this paper we improve existing results and show that with high probability, the game chromatic number χg(Gn,p) of dense random graphs with p ≥ e-o(log n) is asymptotically twice as large as the ordinary chromatic number χ(Gn,p).

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

scholarly journals The game chromatic number of random graphs

2008 ◽  
Vol 32 (2) ◽  
pp. 223-235 ◽  
Author(s):  
Tom Bohman ◽  
Alan Frieze ◽  
Benny Sudakov
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Game Chromatic Number

scholarly journals The eternal game chromatic number of random graphs

2021 ◽  
Vol 95 ◽  
pp. 103324
Author(s):  
Vojtěch Dvořák ◽  
Rebekah Herrman ◽  
Peter van Hintum
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Game Chromatic Number

scholarly journals All finite sets are Ramsey in the maximum norm

2021 ◽  
Vol 9 ◽  
Author(s):  
Andrey Kupavskii ◽  
Arsenii Sagdeev
Keyword(s):  
Chromatic Number ◽  
Metric Space ◽  
Metric Spaces ◽  
Upper Bounds ◽  
Maximum Norm ◽  
Lower And Upper Bounds ◽  
Finite Sets ◽  
Finite Metric Space ◽  
Monochromatic Copy

Abstract For two metric spaces $\mathbb X$ and $\mathcal Y$ the chromatic number $\chi ({{\mathbb X}};{{\mathcal{Y}}})$ of $\mathbb X$ with forbidden $\mathcal Y$ is the smallest k such that there is a colouring of the points of $\mathbb X$ with k colors that contains no monochromatic copy of $\mathcal Y$ . In this article, we show that for each finite metric space $\mathcal {M}$ that contains at least two points the value $\chi \left ({{\mathbb R}}^n_\infty; \mathcal M \right )$ grows exponentially with n. We also provide explicit lower and upper bounds for some special $\mathcal M$ .

scholarly journals Some Extremal Graphs with Respect to Sombor Index

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1202
Author(s):  
Kinkar Chandra Das ◽  
Yilun Shang
Keyword(s):  
Molecular Structure ◽  
Chromatic Number ◽  
Clique Number ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Graph Parameters ◽  
Structure Descriptor

Let G be a graph with set of vertices V(G)(|V(G)|=n) and edge set E(G). Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by SO(G) and is defined as SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) is the degree of the vertex vi in G. In this paper we present some lower and upper bounds on the Sombor index of graph G in terms of graph parameters (clique number, chromatic number, number of pendant vertices, etc.) and characterize the extremal graphs.

scholarly journals Client-Waiter Games on Complete and Random Graphs

10.37236/6039 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Oren Dean ◽  
Michael Krivelevich
Keyword(s):  
Positive Integer ◽  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Upper Bounds ◽  
Connected Component ◽  
Lower And Upper Bounds ◽  
Large Connected Component ◽  
Graph Property ◽  
Player Game

For a graph $ G $, a monotone increasing graph property $ \mathcal{P} $ and positive integer $ q $, we define the Client-Waiter game to be a two-player game which runs as follows. In each turn Waiter is offering Client a subset of at least one and at most $ q+1 $ unclaimed edges of $ G $ from which Client claims one, and the rest are claimed by Waiter. The game ends when all the edges have been claimed. If Client's graph has property $ \mathcal{P} $ by the end of the game, then he wins the game, otherwise Waiter is the winner. In this paper we study several Client-Waiter games on the edge set of the complete graph, and the $ H $-game on the edge set of the random graph. For the complete graph we consider games where Client tries to build a large star, a long path and a large connected component. We obtain lower and upper bounds on the critical bias for these games and compare them with the corresponding Waiter-Client games and with the probabilistic intuition. For the $ H $-game on the random graph we show that the known results for the corresponding Maker-Breaker game are essentially the same for the Client-Waiter game, and we extend those results for the biased games and for trees.

scholarly journals The Connectivity of a Bipartite Graph and Its Bipartite Complementary Graph

2020 ◽  
Vol 30 (03) ◽  
pp. 2040005
Author(s):  
Yingzhi Tian ◽  
Huaping Ma ◽  
Liyun Wu
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Type Inequality ◽  
Bipartite Graphs ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Complementary Graph

In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, any bound on the sum and/or the product of an invariant in a graph [Formula: see text] and the same invariant in the complement [Formula: see text] of [Formula: see text] is called a Nordhaus-Gaddum type inequality or relation. The Nordhaus-Gaddum type inequalities for connectivity have been studied by several authors. For a bipartite graph [Formula: see text] with bipartition ([Formula: see text]), its bipartite complementary graph [Formula: see text] is a bipartite graph with [Formula: see text] and [Formula: see text] and [Formula: see text]. In this paper, we obtain the Nordhaus-Gaddum type inequalities for connectivity of bipartite graphs and its bipartite complementary graphs. Furthermore, we prove that these inequalities are best possible.

MINIMIZING CONGESTION OF LAYOUTS FOR ATM NETWORKS WITH FAULTY LINKS

1999 ◽  
Vol 10 (04) ◽  
pp. 503-512 ◽  
Author(s):  
LESZEK GASIENIEC ◽  
EVANGELOS KRANAKIS ◽  
DANNY KRIZANC ◽  
ANDREZEJ PELC
Keyword(s):  
Lower Bounds ◽  
High Probability ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Atm Networks ◽  
Worst Case ◽  
Precise Analysis ◽  
Order Of Magnitude ◽  
Bounded Size ◽  
Fault Distribution

We consider the problem of constructing virtual path layouts for an ATM network consisting of a complete network Kn of n processors in which a certain number of links may fail. Our main goal is to construct layouts which tolerate any configuration of up to f faults and have the least possible congestion. First, we study the minimal congestion of 1-hop f-tolerant layouts in Kn. For any positive integer f we give upper and lower bounds on this minimal congestion and construct f-tolerant layouts with congestion corresponding to the upper bounds. Our results are based on a precise analysis of the diameter of the network Kn[ℱ] which results from Kn by deleting links from a set ℱ of bounded size. Next we study the minimal congestion of h-hop f-tolerant layouts in Kn, for larger values of the number h of hops. We give upper and lower bounds on the order of magnitude of this congestion, based on results for 1-hop layouts. Finally, we consider a random, rather than worst case, fault distribution where links fail independently with constant probability p<1. Our goal now is to construct layouts with low congestion that tolerate the existing faults with high probability. For any p<1, we show the existence of 1-hop layouts in Kn, with congestion O( log n).

scholarly journals Locating-chromatic number of the edge-amalgamation of trees

2020 ◽  
Vol 4 (2) ◽  
pp. 126
Author(s):  
Dian Kastika Syofyan ◽  
Edy Tri Baskoro ◽  
Hilda Assiyatun
Keyword(s):  
Chromatic Number ◽  
Upper Bounds ◽  
Title Page ◽  
Metric Dimension ◽  
Chromatic Numbers ◽  
Lower And Upper Bounds ◽  
Vertex Set ◽  
Partition Dimension ◽  
Special Case

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>The investigation on the locating-chromatic number of a graph was initiated by Chartrand </span><span>et al. </span><span>(2002). This concept is in fact a special case of the partition dimension of a graph. This topic has received much attention. However, the results are still far from satisfaction. We can define the locating-chromatic number of a graph </span><span>G </span><span>as the smallest integer </span><span>k </span><span>such that there exists a </span><span>k</span><span>-partition of the vertex-set of </span><span>G </span><span>such that all vertices have distinct coordinates with respect to this partition. As we know that the metric dimension of a tree is completely solved. However, the locating-chromatic numbers for most of trees are still open. For </span><span><em>i</em> </span><span>= 1</span><span>, </span><span>2</span><span>, . . . , <em>t</em>, </span><span>let </span><em>T</em><span>i </span><span>be a tree with a fixed edge </span><span>e</span><span>o</span><span>i </span><span>called the terminal edge. The edge-amalgamation of all </span><span>T</span><span>i</span><span>s </span><span>denoted by Edge-Amal</span><span>{</span><span>T</span><span>i</span><span>;</span><span>e</span><span>o</span><span>i</span><span>} </span><span>is a tree formed by taking all the </span><span>T</span><span>i</span><span>s and identifying their terminal edges. In this paper, we study the locating-chromatic number of the edge-amalgamation of arbitrary trees. We give lower and upper bounds for their locating-chromatic numbers and show that the bounds are tight.</span></p></div></div></div>

scholarly journals Stochastic Online Learning with Probabilistic Graph Feedback

2020 ◽  
Vol 34 (04) ◽  
pp. 4675-4682
Author(s):  
Shuai Li ◽  
Wei Chen ◽  
Zheng Wen ◽  
Kwong-Sak Leung
Keyword(s):  
Online Learning ◽  
Lower Bounds ◽  
Random Graphs ◽  
High Probability ◽  
Upper Bounds ◽  
Directed Edge ◽  
Probabilistic Graph ◽  
One Step ◽  
The One ◽  
Independent Probability

We consider a problem of stochastic online learning with general probabilistic graph feedback, where each directed edge in the feedback graph has probability pij. Two cases are covered. (a) The one-step case, where after playing arm i the learner observes a sample reward feedback of arm j with independent probability pij. (b) The cascade case where after playing arm i the learner observes feedback of all arms j in a probabilistic cascade starting from i – for each (i,j) with probability pij, if arm i is played or observed, then a reward sample of arm j would be observed with independent probability pij. Previous works mainly focus on deterministic graphs which corresponds to one-step case with pij ∈ {0,1}, an adversarial sequence of graphs with certain topology guarantees, or a specific type of random graphs. We analyze the asymptotic lower bounds and design algorithms in both cases. The regret upper bounds of the algorithms match the lower bounds with high probability.

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