134 research outputs found
Bipartite, Size, and Online Ramsey Numbers of Some Cycles and Paths
The basic premise of Ramsey Theory states that in a sufficiently large system, complete disorder is impossible. One instance from the world of graph theory says that given two fixed graphs F and H, there exists a finitely large graph G such that any red/blue edge coloring of the edges of G will produce a red copy of F or a blue copy of H. Much research has been conducted in recent decades on quantifying exactly how large G must be if we consider different classes of graphs for F and H. In this thesis, we explore several Ramsey- type problems with a particular focus on paths and cycles. We first examine the bipartite size Ramsey number of a path on n vertices, bˆr(Pn), and give an upper bound using a random graph construction motivated by prior upper bound improvements in similar problems. Next, we consider the size Ramsey number Rˆ (C, Pn) and provide a significant improvement to the upper bound using a very structured graph, the cube of a path, as opposed to a random construction. We also prove a small improvement to the lower bound and show that the r-colored version of this problem is asymptotically linear in rn. Lastly, we give an upper bound for the online Ramsey number R˜ (C, Pn)
Online Ramsey Games in Random Graphs
Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove a lower bound of nβ(F,r) on the typical duration of this game, where β(F,r) is a function that is strictly increasing in r and satisfies limr→∞ β(F,r) = 2 − 1/m2(F), where n2−1/m2(F) is the threshold of the corresponding offline colouring proble
On Metric Ramsey-type Dichotomies
The classical Ramsey theorem, states that every graph contains either a large
clique or a large independent set. Here we investigate similar dichotomic
phenomena in the context of finite metric spaces. Namely, we prove statements
of the form "Every finite metric space contains a large subspace that is nearly
quilateral or far from being equilateral". We consider two distinct
interpretations for being "far from equilateral". Proximity among metric spaces
is quantified through the metric distortion D. We provide tight asymptotic
answers for these problems. In particular, we show that a phase transition
occurs at D=2.Comment: 14 pages, 0 figure
Coloring random graphs online without creating monochromatic subgraphs
Consider the following random process: The vertices of a binomial random
graph are revealed one by one, and at each step only the edges
induced by the already revealed vertices are visible. Our goal is to assign to
each vertex one from a fixed number of available colors immediately and
irrevocably without creating a monochromatic copy of some fixed graph in
the process. Our first main result is that for any and , the threshold
function for this problem is given by , where
denotes the so-called \emph{online vertex-Ramsey density} of
and . This parameter is defined via a purely deterministic two-player game,
in which the random process is replaced by an adversary that is subject to
certain restrictions inherited from the random setting. Our second main result
states that for any and , the online vertex-Ramsey density
is a computable rational number. Our lower bound proof is algorithmic, i.e., we
obtain polynomial-time online algorithms that succeed in coloring as
desired with probability for any .Comment: some minor addition
Ramsey numbers of squares of paths
The Ramsey number R(G;H) has been actively studied for the past 40 years, and it was determined for a large family of pairs (G;H) of graphs. The Ramsey number of paths was determined very early on, but surprisingly very little is known about the Ramsey number for the powers of paths. The r-th power Pr n of a path on n vertices is obtained by joining any two vertices with distance at most r. We determine the exact value of R(P2 n; P2 n) for n large and discuss some related questions
Ramsey-type theorems for metric spaces with applications to online problems
A nearly logarithmic lower bound on the randomized competitive ratio for the
metrical task systems problem is presented. This implies a similar lower bound
for the extensively studied k-server problem. The proof is based on Ramsey-type
theorems for metric spaces, that state that every metric space contains a large
subspace which is approximately a hierarchically well-separated tree (and in
particular an ultrametric). These Ramsey-type theorems may be of independent
interest.Comment: Fix an error in the metadata. 31 pages, 0 figures. Preliminary
version in FOCS '01. To be published in J. Comput. System Sc
Online vertex-coloring games in random graphs
Consider the following one-player game. The vertices of a random graph on n vertices are revealed to the player one by one. In each step, also all edges connecting the newly revealed vertex to preceding vertices are revealed. The player has a fixed number of colors at her disposal, and has to assign one of these to each vertex immediately. However, she is not allowed to create any monochromatic copy of some fixed graph F in the process. For n → ∞, we study how the limiting probability that the player can color all n vertices in this online fashion depends on the edge density of the underlying random graph. For a large family of graphs F, including cliques and cycles of arbitrary size, and any fixed number of colors, we establish explicit threshold functions for this edge density. In particular, we show that the order of magnitude of these threshold functions depends on the number of colors, which is in contrast to the corresponding offline coloring proble
Three results on cycle-wheel Ramsey numbers
Given two graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer N such that, for any graph G of order N, either G1 is a subgraph of G, or G2 is a subgraph of the complement of G. We consider the case that G1 is a cycle and G2 is a (generalized) wheel. We expand the knowledge on exact values of Ramsey numbers in three directions: large cycles versus wheels of odd order; large wheels versus cycles of even order; and large cycles versus generalized odd wheels
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