4,746 research outputs found
Finding combinatorial structures
In this thesis we answer questions in two related areas of combinatorics:
Ramsey theory and asymptotic enumeration.
In Ramsey theory we introduce a new method for finding desired structures.
We find a new upper bound on the Ramsey number of a path against a kth
power of a path.
Using our new method and this result we obtain a new upper bound on the
Ramsey number of the kth power of a long cycle.
As a corollary we show that, while graphs on n vertices with maximum
degree k may in general have Ramsey numbers as large as ckn, if the stronger
restriction that the bandwidth should be at most k is given, then the Ramsey
numbers are bounded by the much smaller value.
We go on to attack an old conjecture of Lehel: by using our new method
we can improve on a result of Luczak, Rodl and Szemeredi [60]. Our new
method replaces their use of the Regularity Lemma, and allows us to prove
that for any n > 218000, whenever the edges of the complete graph on n
vertices are two-coloured there exist disjoint monochromatic cycles covering
all n vertices.
In asymptotic enumeration we examine first the class of bipartite graphs
with some forbidden induced subgraph H. We obtain some results for every
H, with special focus on the cases where the growth speed of the class is
factorial, and make some comments on a connection to clique-width. We
then move on to a detailed discussion of 2-SAT functions. We find the correct
asymptotic formula for the number of 2-SAT functions
on n variables (an improvement on a result of Bollob´as, Brightwell and
Leader [13], who found the dominant term in the exponent), the first error
term for this formula, and some bounds on smaller error terms. Finally
we obtain various expected values in the uniform model of random 2-SAT
functions
Dynamic programming for graphs on surfaces
We provide a framework for the design and analysis of dynamic
programming algorithms for surface-embedded graphs on n vertices
and branchwidth at most k. Our technique applies to general families
of problems where standard dynamic programming runs in 2O(k·log k).
Our approach combines tools from topological graph theory and
analytic combinatorics.Postprint (updated version
Enumeration of graphs with a heavy-tailed degree sequence
In this paper, we asymptotically enumerate graphs with a given degree
sequence d=(d_1,...,d_n) satisfying restrictions designed to permit
heavy-tailed sequences in the sparse case (i.e. where the average degree is
rather small). Our general result requires upper bounds on functions of M_k=
\sum_{i=1}^n [d_i]_k for a few small integers k\ge 1. Note that M_1 is simply
the total degree of the graphs. As special cases, we asymptotically enumerate
graphs with (i) degree sequences satisfying M_2=o(M_1^{ 9/8}); (ii) degree
sequences following a power law with parameter gamma>5/2; (iii) power-law
degree sequences that mimic independent power-law "degrees" with parameter
gamma>1+\sqrt{3}\approx 2.732; (iv) degree sequences following a certain
"long-tailed" power law; (v) certain bi-valued sequences. A previous result on
sparse graphs by McKay and the second author applies to a wide range of degree
sequences but requires Delta =o(M_1^{1/3}), where Delta is the maximum degree.
Our new result applies in some cases when Delta is only barely o(M_1^ {3/5}).
Case (i) above generalises a result of Janson which requires M_2=O(M_1) (and
hence M_1=O(n) and Delta=O(n^{1/2})). Cases (ii) and (iii) provide the first
asymptotic enumeration results applicable to degree sequences of real-world
networks following a power law, for which it has been empirically observed that
2<gamma<3.Comment: 34 page
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