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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
The distribution of height and diameter in random non-plane binary trees
This study is dedicated to precise distributional analyses of the height of
non-plane unlabelled binary trees ("Otter trees"), when trees of a given size
are taken with equal likelihood. The height of a rooted tree of size is
proved to admit a limiting theta distribution, both in a central and local
sense, as well as obey moderate as well as large deviations estimates. The
approximations obtained for height also yield the limiting distribution of the
diameter of unrooted trees. The proofs rely on a precise analysis, in the
complex plane and near singularities, of generating functions associated with
trees of bounded height
Scaling limits of k-ary growing trees
For each integer , we introduce a sequence of -ary discrete
trees constructed recursively by choosing at each step an edge uniformly among
the present edges and grafting on "its middle" new edges. When ,
this corresponds to a well-known algorithm which was first introduced by
R\'emy. Our main result concerns the asymptotic behavior of these trees as
becomes large: for all , the sequence of -ary trees grows at speed
towards a -ary random real tree that belongs to the family of
self-similar fragmentation trees. This convergence is proved with respect to
the Gromov-Hausdorff-Prokhorov topology. We also study embeddings of the
limiting trees when varies
Generalizations of the Tree Packing Conjecture
The Gy\'arf\'as tree packing conjecture asserts that any set of trees with
vertices has an (edge-disjoint) packing into the complete graph
on vertices. Gy\'arf\'as and Lehel proved that the conjecture holds in some
special cases. We address the problem of packing trees into -chromatic
graphs. In particular, we prove that if all but three of the trees are stars
then they have a packing into any -chromatic graph. We also consider several
other generalizations of the conjecture
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