8 research outputs found
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
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
Polynomial bounds for chromatic number VIII. Excluding a path and a complete multipartite graph
We prove that for every path H, and every integer d, there is a polynomial f such that every graph
G with chromatic number greater than f(t) either contains H as an induced subgraph, or contains
as a subgraph the complete d-partite graph with parts of cardinality t. For t = 1 and general d this
is a classical theorem of Gyárfás, and for d = 2 and general t this is a theorem of Bonamy et al
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
Polynomial bounds for chromatic number. I. Excluding a biclique and an induced tree
Let H be a tree. It was proved by Rödl that graphs that do not contain H as an induced subgraph, and do not contain the complete bipartite graph Kt,t as a subgraph, have bounded chromatic number. Kierstead and Penrice strengthened this, showing that such graphs have bounded degeneracy. Here we give a further strengthening, proving that for every tree H, the degeneracy is at most polynomial in t. This answers a question of Bonamy, Bousquet, Pilipczuk, Rzążewski, Thomassé, and Walczak.Engineering and Physical Sciences Research Council (EPSRC), EP/V007327/1 || Air Force Office of Scientific Research (AFOSR), A9550-19-1-0187 || Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2020-0391
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