8,687 research outputs found
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 for partially-ordered sets
We present a refinement of Ramsey numbers by considering graphs with a
partial ordering on their vertices. This is a natural extension of the ordered
Ramsey numbers. We formalize situations in which we can use arbitrary families
of partially-ordered sets to form host graphs for Ramsey problems. We explore
connections to well studied Tur\'an-type problems in partially-ordered sets,
particularly those in the Boolean lattice. We find a strong difference between
Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the
partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl
Complexity of Computing the Anti-Ramsey Numbers for Paths
The anti-Ramsey numbers are a fundamental notion in graph theory, introduced
in 1978, by Erd\" os, Simonovits and S\' os. For given graphs and the
\emph{anti-Ramsey number} is defined to be the maximum
number such that there exists an assignment of colors to the edges of
in which every copy of in has at least two edges with the same
color.
There are works on the computational complexity of the problem when is a
star. Along this line of research, we study the complexity of computing the
anti-Ramsey number , where is a path of length .
First, we observe that when , the problem is hard; hence, the
challenging part is the computational complexity of the problem when is a
fixed constant.
We provide a characterization of the problem for paths of constant length.
Our first main contribution is to prove that computing for
every integer is NP-hard. We obtain this by providing several structural
properties of such coloring in graphs. We investigate further and show that
approximating to a factor of is hard
already in -partite graphs, unless P=NP. We also study the exact complexity
of the precolored version and show that there is no subexponential algorithm
for the problem unless ETH fails for any fixed constant .
Given the hardness of approximation and parametrization of the problem, it is
natural to study the problem on restricted graph families. We introduce the
notion of color connected coloring and employing this structural property. We
obtain a linear time algorithm to compute , for every
integer , when the host graph, , is a tree
Ramsey Goodness and Beyond
In a seminal paper from 1983, Burr and Erdos started the systematic study of
Ramsey numbers of cliques vs. large sparse graphs, raising a number of
problems. In this paper we develop a new approach to such Ramsey problems using
a mix of the Szemeredi regularity lemma, embedding of sparse graphs, Turan type
stability, and other structural results. We give exact Ramsey numbers for
various classes of graphs, solving all but one of the Burr-Erdos problems.Comment: A new reference is adde
Chromatic Ramsey number of acyclic hypergraphs
Suppose that is an acyclic -uniform hypergraph, with . We
define the (-color) chromatic Ramsey number as the smallest
with the following property: if the edges of any -chromatic -uniform
hypergraph are colored with colors in any manner, there is a monochromatic
copy of . We observe that is well defined and where
is the -color Ramsey number of . We give linear upper bounds
for when T is a matching or star, proving that for , and where
and are, respectively, the -uniform matching and star with
edges.
The general bounds are improved for -uniform hypergraphs. We prove that
, extending a special case of Alon-Frankl-Lov\'asz' theorem.
We also prove that , which is sharp for . This is
a corollary of a more general result. We define as the 1-intersection
graph of , whose vertices represent hyperedges and whose edges represent
intersections of hyperedges in exactly one vertex. We prove that for any -uniform hypergraph (assuming ). The proof uses the list coloring version of Brooks' theorem.Comment: 10 page
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