337 research outputs found
The approximate Loebl-Komlos-Sos conjecture and embedding trees in sparse graphs
Loebl, Koml\'os and S\'os conjectured that every -vertex graph with at
least vertices of degree at least contains each tree of order
as a subgraph. We give a sketch of a proof of the approximate version of
this conjecture for large values of .
For our proof, we use a structural decomposition which can be seen as an
analogue of Szemer\'edi's regularity lemma for possibly very sparse graphs.
With this tool, each graph can be decomposed into four parts: a set of vertices
of huge degree, regular pairs (in the sense of the regularity lemma), and two
other objects each exhibiting certain expansion properties. We then exploit the
properties of each of the parts of to embed a given tree .
The purpose of this note is to highlight the key steps of our proof. Details
can be found in [arXiv:1211.3050]
Bounding mean orders of sub--trees of -trees
For a -tree , we prove that the maximum local mean order is attained in
a -clique of degree and that it is not more than twice the global mean
order. We also bound the global mean order if has no -cliques of degree
and prove that for large order, the -star attains the minimum global
mean order. These results solve the remaining problems of Stephens and
Oellermann [J. Graph Theory 88 (2018), 61-79] concerning the mean order of
sub--trees of -trees.Comment: 20 Pages, 6 Figure
The path minimises the average size of a connected induced subgraph
We prove that among all graphs of order n, the path uniquely minimises the
average order of its connected induced subgraphs. This confirms a conjecture of
Kroeker, Mol and Oellermann, and generalises a classical result of Jamison for
trees, as well as giving a new, shorter proof of the latter. While this paper
was being prepared, a different proof was given by Andrew Vince.Comment: 9 pages, 1 figure. Changed title, new figure and minor rewritin
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