349 research outputs found
All partitions have small parts - Gallai-Ramsey numbers of bipartite graphs
Gallai-colorings are edge-colored complete graphs in which there are no
rainbow triangles. Within such colored complete graphs, we consider Ramsey-type
questions, looking for specified monochromatic graphs. In this work, we
consider monochromatic bipartite graphs since the numbers are known to grow
more slowly than for non-bipartite graphs. The main result shows that it
suffices to consider only -colorings which have a special partition of the
vertices. Using this tool, we find several sharp numbers and conjecture the
sharp value for all bipartite graphs. In particular, we determine the
Gallai-Ramsey numbers for all bipartite graphs with two vertices in one part
and initiate the study of linear forests
The typical structure of Gallai colorings and their extremal graphs
An edge coloring of a graph is a Gallai coloring if it contains no
rainbow triangle. We show that the number of Gallai -colorings of is
. This result indicates that
almost all Gallai -colorings of use only 2 colors. We also study the
extremal behavior of Gallai -colorings among all -vertex graphs. We prove
that the complete graph admits the largest number of Gallai -colorings
among all -vertex graphs when is sufficiently large, while for , it is the complete bipartite graph . Our main approach is based on the hypergraph container method,
developed independently by Balogh, Morris, and Samotij as well as by Saxton and
Thomason, together with some stability results for containers.Comment: 28 page
Brooks's theorem for measurable colorings
We generalize Brooks's theorem to show that if is a Borel graph on a
standard Borel space of degree bounded by which contains no
-cliques, then admits a -measurable -coloring with respect
to any Borel probability measure on , and a Baire measurable
-coloring with respect to any compatible Polish topology on . The proof
of this theorem uses a new technique for constructing one-ended spanning
subforests of Borel graphs, as well as ideas from the study of list colorings.
We apply the theorem to graphs arising from group actions to obtain factor of
IID -colorings of Cayley graphs of degree , except in two exceptional
cases.Comment: Minor correction
A conjecture on Gallai-Ramsey numbers of even cycles and paths
A Gallai coloring is a coloring of the edges of a complete graph without
rainbow triangles, and a Gallai -coloring is a Gallai coloring that uses at
most colors. Given an integer and graphs , the
Gallai-Ramsey number is the least integer such that
every Gallai -coloring of the complete graph contains a monochromatic
copy of in color for some . When , we simply write . We study Gallai-Ramsey numbers of
even cycles and paths. For all and , let be a
path on vertices for all and
. Let for all
with . The first author
recently conjectured that . The truth of this conjecture implies that
for all and , and
for all and . In this paper, we
prove that the aforementioned conjecture holds for and all
. Our proof relies only on Gallai's result and the classical Ramsey
numbers , where . We
believe the recoloring method we developed here will be very useful for solving
subsequent cases, and perhaps the conjecture
List colorings of -minor-free graphs with special list assignments
A {\it list assignment} of a graph is a function that assigns a set
(list) of colors to every vertex of . Graph is called {\it
-list colorable} if it admits a vertex coloring such that for all and for all .
The following question was raised by Bruce Richter. Let be a planar,
3-connected graph that is not a complete graph. Denoting by the degree
of vertex , is -list colorable for every list assignment with
for all ?
More generally, we ask for which pairs the following question has an
affirmative answer. Let and be integers and let be a
-minor-free -connected graph that is not a Gallai tree (i.e., at least
one block of is neither a complete graph nor an odd cycle). Is -list
colorable for every list assignment with for all
?
We investigate this question by considering the components of , where
is the set of vertices with small degree in .
We are especially interested in the minimum distance in between
the components of .Comment: 14 pages, 2 figures, 2 table
Gallai-Ramsey number of an 8-cycle
Given graphs and and a positive integer , the Gallai-Ramsey number
is the minimum integer such that for any integer , every -edge-coloring of contains either a rainbow copy of or
a monochromatic copy of . These numbers have recently been studied for the
case when , where still only a few precise numbers are known for all
. In this paper, we extend the known precise Gallai-Ramsey numbers to
include for all
A Brooks type theorem for the maximum local edge connectivity
For a graph , let \cn(G) and \la(G) denote the chromatic number of
and the maximum local edge connectivity of , respectively. A result of Dirac
\cite{Dirac53} implies that every graph satisfies \cn(G)\leq \la(G)+1. In
this paper we characterize the graphs for which \cn(G)=\la(G)+1. The case
\la(G)=3 was already solved by Alboulker {\em et al.\,} \cite{AlboukerV2016}.
We show that a graph with \la(G)=k\geq 4 satisfies \cn(G)=k+1 if and
only if contains a block which can be obtained from copies of by
repeated applications of the Haj\'os join.Comment: 15 pages, 1 figur
Multicolor Gallai-Ramsey numbers of and
A Gallai coloring is a coloring of the edges of a complete graph without
rainbow triangles, and a Gallai -coloring is a Gallai coloring that uses
colors. We study Ramsey-type problems in Gallai colorings. Given an integer
and a graph , the Gallai-Ramsey number is the least
positive integer such that every Gallai -coloring of the complete graph
on vertices contains a monochromatic copy of . It turns out that
is more well-behaved than the classical Ramsey number .
However, finding exact values of is far from trivial. In this paper,
we study Gallai-Ramsey numbers of odd cycles. We prove that for
and all , . This new result provides
partial evidence for the first two open cases of the Triple Odd Cycle
Conjecture of Bondy and Erd\H{o}s from 1973. Our technique relies heavily on
the structural result of Gallai on Gallai colorings of complete graphs. We
believe the method we developed can be used to determine the exact values of
for all .Comment: A long and technical proof of Gallai-Ramsey numbers of C9 can be
found in our preprint arXiv:1709.06130, which will not be submitted for
publicatio
Gallai-Ramsey numbers of with multiple colors
We study Ramsey-type problems in Gallai-colorings. Given a graph and an
integer , the Gallai-Ramsey number is the least positive
integer such that every -coloring of the edges of the complete graph on
vertices contains either a rainbow triangle or a monochromatic copy of .
It turns out that behaves more nicely than the classical Ramsey
number . However, finding exact values of is far from
trivial. In this paper, we prove that for all
. This new result provides partial evidence for the first open case of
the Triple Odd Cycle Conjecture of Bondy and Erd\H{o}s from 1973. Our technique
relies heavily on the structural result of Gallai on edge-colorings of complete
graphs without rainbow triangles. We believe the method we developed can be
used to determine the exact values of for odd integers
.Comment: 15 pages, 3 figures, one overlooked case, namely Claim 2.10, was
adde
Gallai colorings and domination in multipartite digraphs
Assume that D is a digraph without cyclic triangles and its vertices are
partitioned into classes A_1,...,A_t of independent vertices. A set
is called a dominating set of size |S| if for any vertex
there is a w in U such that (w,v) is in E(D). Let
beta(D) be the cardinality of the largest independent set of D whose vertices
are from different partite classes of D. Our main result says that there exists
a h=h(beta(D)) such that D has a dominating set of size at most h. This result
is applied to settle a problem related to generalized Gallai colorings, edge
colorings of graphs without 3-colored triangles.Comment: 15 pages, 8 figure
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