218 research outputs found
Gallai-Ramsey Number for Classes of Brooms
Given a graph , we consider the problem of finding the minimum number such that any edge colored complete graph on vertices contains either a rainbow colored triangle or a monochromatic copy of the graph , denoted . More precisely we consider where is a broom graph with representing the number of vertices on the handle and representing the number of bristle vertices. We develop a technique to reduce the difficulty of finding , and use the technique to prove a few cases with a fixed handle length, but arbitrarily many bristles. Further, we find upper and lower bounds for any broom
Size Ramsey Numbers Involving Double Stars and Brooms
The topics of this thesis lie in graph Ramsey theory. Given two graphs G and H, by the Ramsey theorem, there exist infinitely many graphs F such that if we partition the edges of F into two sets, say Red and Blue, then either the graph induced by the red edges contains G or the graph induced by the blue edges contains H. The minimum order of F is called the Ramsey number and the minimum of the size of F is called the size Ramsey number. They are denoted by r(G, H) and Λr(G, H), respectively. We will investigate size Ramsey numbers involving double stars and brooms
Induced subgraphs of graphs with large chromatic number. XIII. New brooms
Gy\'arf\'as and Sumner independently conjectured that for every tree , the
class of graphs not containing as an induced subgraph is -bounded,
that is, the chromatic numbers of graphs in this class are bounded above by a
function of their clique numbers. This remains open for general trees , but
has been proved for some particular trees. For , let us say a broom of
length is a tree obtained from a -edge path with ends by adding
some number of leaves adjacent to , and we call its handle. A tree
obtained from brooms of lengths by identifying their handles is a
-multibroom. Kierstead and Penrice proved that every
-multibroom satisfies the Gy\'arf\'as-Sumner conjecture, and
Kierstead and Zhu proved the same for -multibrooms. In this paper
give a common generalization: we prove that every
-multibroom satisfies the Gy\'arf\'as-Sumner conjecture
Induced subgraphs of graphs with large chromatic number. XI. Orientations
Fix an oriented graph H, and let G be a graph with bounded clique number and
very large chromatic number. If we somehow orient its edges, must there be an
induced subdigraph isomorphic to H? Kierstead and Rodl raised this question for
two specific kinds of digraph H: the three-edge path, with the first and last
edges both directed towards the interior; and stars (with many edges directed
out and many directed in). Aboulker et al subsequently conjectured that the
answer is affirmative in both cases. We give affirmative answers to both
questions
Polynomial -binding functions for -broom-free graphs
For any positive integer , a \emph{-broom} is a graph obtained from
by subdividing an edge once. In this paper, we show that, for
graphs without induced -brooms, we have ,
where and are the chromatic number and clique number of
, respectively. When , this answers a question of Schiermeyer and
Randerath. Moreover, for , we strengthen the bound on to
, confirming a conjecture of Sivaraman. For and
\{-broom, \}-free graphs, we improve the bound to
.Comment: 14 pages, 1 figur
Induced subgraphs of graphs with large chromatic number. XII. Distant stars
The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph
with bounded clique number and very large chromatic number contains H as an
induced subgraph. This is still open, although it has been proved for a few
simple families of trees, including trees of radius two, some special trees of
radius three, and subdivided stars. These trees all have the property that
their vertices of degree more than two are clustered quite closely together. In
this paper, we prove the conjecture for two families of trees which do not have
this restriction. As special cases, these families contain all double-ended
brooms and two-legged caterpillars
A survey of -boundedness
If a graph has bounded clique number, and sufficiently large chromatic
number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made
a number of challenging conjectures about this in the early 1980's, which have
remained open until recently; but in the last few years there has been
substantial progress. This is a survey of where we are now
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