64 research outputs found
Ramsey and Tur\'an numbers of sparse hypergraphs
Degeneracy plays an important role in understanding Tur\'an- and Ramsey-type
properties of graphs. Unfortunately, the usual hypergraphical generalization of
degeneracy fails to capture these properties. We define the skeletal degeneracy
of a -uniform hypergraph as the degeneracy of its -skeleton (i.e., the
graph formed by replacing every -edge by a -clique). We prove that
skeletal degeneracy controls hypergraph Tur\'an and Ramsey numbers in a similar
manner to (graphical) degeneracy.
Specifically, we show that -uniform hypergraphs with bounded skeletal
degeneracy have linear Ramsey number. This is the hypergraph analogue of the
Burr-Erd\H{o}s conjecture (proved by Lee). In addition, we give upper and lower
bounds of the same shape for the Tur\'an number of a -uniform -partite
hypergraph in terms of its skeletal degeneracy. The proofs of both results use
the technique of dependent random choice. In addition, the proof of our Ramsey
result uses the `random greedy process' introduced by Lee in his resolution of
the Burr-Erd\H{o}s conjecture.Comment: 33 page
The size-Ramsey number of powers of paths
Given graphs and and a positive integer , say that
\emph{is -Ramsey for} , denoted
, if every -colouring of the edges of
contains a monochromatic copy of . The \emph{size-Ramsey number} \sr(H) of
a graph is defined to be
\sr(H)=\min\{|E(G)|\colon G\rightarrow (H)_2\}. Answering a
question of Conlon, we prove that, for every fixed~, we have
\sr(P_n^k)=O(n), where~ is the th power of the
-vertex path (i.e., the graph with vertex set and
all edges such that the distance between and in
is at most ). Our proof is probabilistic, but can also be made constructive.Most of the work for this paper was done during my PhD, which was half funded by EPSRC grant reference 1360036, and half by Merton College Oxford.
The third author was partially supported by FAPESP
(Proc.~2013/03447-6) and by CNPq (Proc.~459335/2014-6,
310974/2013-5). The fifth author was
supported by FAPESP (Proc.~2013/11431-2, Proc.~2013/03447-6 and
Proc.~2018/04876-1) and partially by CNPq (Proc.~459335/2014-6).
This research was supported in part by CAPES (Finance Code 001).
The collaboration of part of the authors was supported by a
CAPES/DAAD PROBRAL grant (Proc.~430/15)
Generation of Graph Classes with Efficient Isomorph Rejection
In this thesis, efficient isomorph-free generation of graph classes with the method of
generation by canonical construction path(GCCP) is discussed. The method GCCP
has been invented by McKay in the 1980s. It is a general method to recursively generate
combinatorial objects avoiding isomorphic copies. In the introduction chapter, the
method of GCCP is discussed and is compared to other well-known methods of generation.
The generation of the class of quartic graphs is used as an example to explain
this method. Quartic graphs are simple regular graphs of degree four. The programs,
we developed based on GCCP, generate quartic graphs with 18 vertices more than two
times as efficiently as the well-known software GENREG does.
This thesis also demonstrates how the class of principal graph pairs can be generated
exhaustively in an efficient way using the method of GCCP. The definition and
importance of principal graph pairs come from the theory of subfactors where each
subfactor can be modelled as a principal graph pair. The theory of subfactors has
applications in the theory of von Neumann algebras, operator algebras, quantum algebras
and Knot theory as well as in design of quantum computers. While it was
initially expected that the classification at index 3 + â5 would be very complicated,
using GCCP to exhaustively generate principal graph pairs was critical in completing
the classification of small index subfactors to index 5Œ.
The other set of classes of graphs considered in this thesis contains graphs without
a given set of cycles. For a given set of graphs, H, the TurĂĄn Number of H, ex(n,H),
is defined to be the maximum number of edges in a graph on n vertices without a
subgraph isomorphic to any graph in H. Denote by EX(n,H), the set of all extremal
graphs with respect to n and H, i.e., graphs with n vertices, ex(n,H) edges and no
subgraph isomorphic to any graph in H. We consider this problem when H is a set of
cycles. New results for ex(n, C) and EX(n, C) are introduced using a set of algorithms
based on the method of GCCP. Let K be an arbitrary subset of {C3, C4, C5, . . . , C32}.
For given n and a set of cycles, C, these algorithms can be used to calculate ex(n, C)
and extremal graphs in Ex(n, C) by recursively extending smaller graphs without any
cycle in C where C = K or C = {C3, C5, C7, . . .} ᎠK and nâ€64. These results are
considerably in excess of the previous results of the many researchers who worked on
similar problems. In the last chapter, a new class of canonical relabellings for graphs, hierarchical
canonical labelling, is introduced in which if the vertices of a graph, G, is canonically
labelled by {1, . . . , n}, then G\{n} is also canonically labelled. An efficient hierarchical
canonical labelling is presented and the application of this labelling in generation
of combinatorial objects is discussed
The multicolour size-Ramsey number of powers of paths
Given a positive integer s, a graph G is s-Ramsey for a graph H, denoted Gâ(H)s, if every s-colouring of the edges of G contains a monochromatic copy of H. The s-colour size-Ramsey number rËs(H) of a graph H is defined to be rËs(H)=minâĄ{|E(G)|:Gâ(H)s}. We prove that, for all positive integers k and s, we have rËs(Pnk)=O(n), where Pnk is the kth power of the n-vertex path Pn
On the Voting Time of the Deterministic Majority Process
In the deterministic binary majority process we are given a simple graph
where each node has one out of two initial opinions. In every round, every node
adopts the majority opinion among its neighbors. By using a potential argument
first discovered by Goles and Olivos (1980), it is known that this process
always converges in rounds to a two-periodic state in which every node
either keeps its opinion or changes it in every round.
It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the
bound on the convergence time of the deterministic binary majority
process is indeed tight even for dense graphs. However, in many graphs such as
the complete graph, from any initial opinion assignment, the process converges
in just a constant number of rounds.
By carefully exploiting the structure of the potential function by Goles and
Olivos (1980), we derive a new upper bound on the convergence time of the
deterministic binary majority process that accounts for such exceptional cases.
We show that it is possible to identify certain modules of a graph in order
to obtain a new graph with the property that the worst-case
convergence time of is an upper bound on that of . Moreover, even
though our upper bound can be computed in linear time, we show that, given an
integer , it is NP-hard to decide whether there exists an initial opinion
assignment for which it takes more than rounds to converge to the
two-periodic state.Comment: full version of brief announcement accepted at DISC'1
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