56,295 research outputs found
On the maximum order of graphs embedded in surfaces
The maximum number of vertices in a graph of maximum degree and
fixed diameter is upper bounded by . If we
restrict our graphs to certain classes, better upper bounds are known. For
instance, for the class of trees there is an upper bound of
for a fixed . The main result of
this paper is that graphs embedded in surfaces of bounded Euler genus
behave like trees, in the sense that, for large , such graphs have
orders bounded from above by begin{cases} c(g+1)(\Delta-1)^{\lfloor
k/2\rfloor} & \text{if $k$ is even} c(g^{3/2}+1)(\Delta-1)^{\lfloor k/2\rfloor}
& \text{if $k$ is odd}, \{cases} where is an absolute constant. This
result represents a qualitative improvement over all previous results, even for
planar graphs of odd diameter . With respect to lower bounds, we construct
graphs of Euler genus , odd diameter , and order
for some absolute constant
. Our results answer in the negative a question of Miller and
\v{S}ir\'a\v{n} (2005).Comment: 13 pages, 3 figure
Untangling polygons and graphs
Untangling is a process in which some vertices of a planar graph are moved to
obtain a straight-line plane drawing. The aim is to move as few vertices as
possible. We present an algorithm that untangles the cycle graph C_n while
keeping at least \Omega(n^{2/3}) vertices fixed. For any graph G, we also
present an upper bound on the number of fixed vertices in the worst case. The
bound is a function of the number of vertices, maximum degree and diameter of
G. One of its consequences is the upper bound O((n log n)^{2/3}) for all
3-vertex-connected planar graphs.Comment: 11 pages, 3 figure
On the diameter of random planar graphs
We show that the diameter D(G_n) of a random labelled connected planar graph
with n vertices is equal to n^{1/4+o(1)}, in probability. More precisely there
exists a constant c>0 such that the probability that D(G_n) lies in the
interval (n^{1/4-\epsilon},n^{1/4+\epsilon}) is greater than
1-\exp(-n^{c\epsilon}) for {\epsilon} small enough and n>n_0(\epsilon). We
prove similar statements for 2-connected and 3-connected planar graphs and
maps.Comment: 24 pages, 7 figure
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
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