95 research outputs found
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
A survey on constructive methods for the Oberwolfach problem and its variants
The generalized Oberwolfach problem asks for a decomposition of a graph
into specified 2-regular spanning subgraphs , called factors.
The classic Oberwolfach problem corresponds to the case when all of the factors
are pairwise isomorphic, and is the complete graph of odd order or the
complete graph of even order with the edges of a -factor removed. When there
are two possible factor types, it is called the Hamilton-Waterloo problem.
In this paper we present a survey of constructive methods which have allowed
recent progress in this area. Specifically, we consider blow-up type
constructions, particularly as applied to the case when each factor consists of
cycles of the same length. We consider the case when the factors are all
bipartite (and hence consist of even cycles) and a method for using circulant
graphs to find solutions. We also consider constructions which yield solutions
with well-behaved automorphisms.Comment: To be published in the Fields Institute Communications book series.
23 pages, 2 figure
Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking
We propose a new camera-based method of robot identification, tracking and
orientation estimation. The system utilises coloured lights mounted in a circle
around each robot to create unique colour sequences that are observed by a
camera. The number of robots that can be uniquely identified is limited by the
number of colours available, , the number of lights on each robot, , and
the number of consecutive lights the camera can see, . For a given set of
parameters, we would like to maximise the number of robots that we can use. We
model this as a combinatorial problem and show that it is equivalent to finding
the maximum number of disjoint -cycles in the de Bruijn graph
.
We provide several existence results that give the maximum number of cycles
in in various cases. For example, we give an optimal
solution when . Another construction yields many cycles in larger
de Bruijn graphs using cycles from smaller de Bruijn graphs: if
can be partitioned into -cycles, then
can be partitioned into -cycles for any divisor of
. The methods used are based on finite field algebra and the combinatorics
of words.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied
Mathematic
Kirchhoff Graphs
Kirchhoff\u27s laws are well-studied for electrical networks with voltage and current sources, and edges marked by resistors. Kirchhoff\u27s voltage law states that the sum of voltages around any circuit of the network graph is zero, while Kirchhoff\u27s current law states that the sum of the currents along any cutset of the network graph is zero. Given a network, these requirements may be encoded by the circuit matrix and cutset matrix of the network graph. The columns of these matrices are indexed by the edges of the network graph, and their row spaces are orthogonal complements. For (chemical or electrochemical) reaction networks, one must naturally study the opposite problem, beginning with the stoichiometric matrix rather than the network graph. This leads to the following question: given such a matrix, what is a suitable graphic rendering of a network that properly visualizes the underlying chemical reactions? Although we can not expect uniqueness, the goal is to prove existence of such a graph for any matrix. Specifically, we study Kirchhoff graphs, originally introduced by Fehribach. Mathematically, Kirchhoff graphs represent the orthocomplementarity of the row space and null space of integer-valued matrices. After introducing the definition of Kirchhoff graphs, we will survey Kirchhoff graphs in the context of several diverse branches of mathematics. Beginning with combinatorial group theory, we consider Cayley graphs of the additive group of vector spaces, and resolve the existence problem for matrices over finite fields. Moving to linear algebra, we draw a number of conclusions based on a purely matrix-theoretic definition of Kirchhoff graphs, specifically regarding the number of vector edges. Next we observe a geometric approach, reviewing James Clerk Maxwell\u27s theory of reciprocal figures, and presenting a number of results on Kirchhoff duality. We then turn to algebraic combinatorics, where we study equitable partitions, quotients, and graph automorphisms. In addition to classifying the matrices that are the quotient of an equitable partition, we demonstrate that many Kirchhoff graphs arise from equitable edge-partitions of directed graphs. Finally we study matroids, where we review Tutte\u27s algorithm for determining when a binary matroid is graphic, and extend this method to show that every binary matroid is Kirchhoff. The underlying theme throughout each of these investigations is determining new ways to both recognize and construct Kirchhoff graphs
Minimal Ramsey graphs, orthogonal Latin squares, and hyperplane coverings
This thesis consists of three independent parts.
The first part of the thesis is concerned with Ramsey theory. Given an integer , a graph is said to be \emph{-Ramsey} for another graph if in any -edge-coloring of there exists a monochromatic copy of . The central line of research in this area investigates the smallest number of vertices in a -Ramsey graph for a given . In this thesis, we explore two different directions. First, we will be interested in the smallest possible minimum degree of a minimal (with respect to subgraph inclusion) -Ramsey graph for a given . This line of research was initiated by Burr, Erdős, and Lovász in the 1970s. We study the minimum degree of a minimal Ramsey graph for a random graph and investigate how many vertices of small degree a minimal Ramsey graph for a given can contain. We also consider the minimum degree problem in a more general asymmetric setting. Second, it is interesting to ask how small modifications to the graph affect the corresponding collection of -Ramsey graphs. Building upon the work of Fox, Grinshpun, Liebenau, Person, and Szabó and Rödl and Siggers, we prove that adding even a single pendent edge to the complete graph changes the collection of 2-Ramsey graphs significantly.
The second part of the thesis deals with orthogonal Latin squares. A {\em Latin square of order } is an array with entries in such that each integer appears exactly once in every row and every column. Two Latin squares and are said to be {\em orthogonal} if, for all , there is a unique pair such that and ; a system of {\em mutually orthogonal Latin squares}, or a {\em -MOLS}, is a set of pairwise orthogonal Latin squares. Motivated by a well-known result determining the number of different Latin squares of order log-asymptotically, we study the number of -MOLS of order . Earlier results on this problem were obtained by Donovan and Grannell and Keevash and Luria. We establish new upper bounds for a wide range of values of . We also prove a new, log-asymptotically tight, bound on the maximum number of other squares a single Latin square can be orthogonal to.
The third part of the thesis is concerned with grid coverings with multiplicities. In particular, we study the minimum number of hyperplanes necessary to cover all points but one of a given finite grid at least times, while covering the remaining point fewer times. We study this problem for the grid , determining the number exactly when one of the parameters and is much larger than the other and asymptotically in all other cases. This generalizes a classic result of Jamison for . Additionally, motivated by the recent work of Clifton and Huang and Sauermann and Wigderson for the hypercube , we study hyperplane coverings for different grids over , under the stricter condition that the remaining point is omitted completely. We focus on two-dimensional real grids, showing a variety of results and demonstrating that already this setting offers a range of possible behaviors.Diese Dissertation besteht aus drei unabh\"angigen Teilen.
Der erste Teil beschäftigt sich mit Ramseytheorie. Für eine ganze Zahl nennt man einen Graphen \emph{-Ramsey} f\"ur einen anderen Graphen , wenn jede Kantenf\"arbung mit Farben einen einfarbigen Teilgraphen enthält, der isomorph zu ist. Das zentrale Problem in diesem Gebiet ist die minimale Anzahl von Knoten in einem solchen Graphen zu bestimmen. In dieser Dissertation betrachten wir zwei verschiedene Varianten. Als erstes, beschäftigen wir uns mit dem kleinstm\"oglichen Minimalgrad eines minimalen (bezüglich Teilgraphen) -Ramsey-Graphen f\"ur einen gegebenen Graphen . Diese Frage wurde zuerst von Burr, Erd\H{o}s und Lov\'asz in den 1970er-Jahren studiert. Wir betrachten dieses Problem f\"ur einen Zufallsgraphen und untersuchen, wie viele Knoten kleinen Grades ein Ramsey-Graph f\"ur gegebenes enthalten kann. Wir untersuchen auch eine asymmetrische Verallgemeinerung des Minimalgradproblems. Als zweites betrachten wir die Frage, wie sich die Menge aller -Ramsey-Graphen f\"ur verändert, wenn wir den Graphen modifizieren. Aufbauend auf den Arbeiten von Fox, Grinshpun, Liebenau, Person und Szabó und Rödl und Siggers beweisen wir, dass bereits der Graph, der aus mit einer h\"angenden Kante besteht, eine sehr unterschiedliche Menge von 2-Ramsey-Graphen besitzt im Vergleich zu .
Im zweiten Teil geht es um orthogonale lateinische Quadrate. Ein \emph{lateinisches Quadrat der Ordnung } ist eine -Matrix, gef\"ullt mit den Zahlen aus , in der jede Zahl genau einmal pro Zeile und einmal pro Spalte auftritt. Zwei lateinische Quadrate sind \emph{orthogonal} zueinander, wenn f\"ur alle genau ein Paar existiert, sodass es und gilt. Ein \emph{k-MOLS der Ordnung } ist eine Menge von lateinischen Quadraten, die paarweise orthogonal sind. Motiviert von einem bekannten Resultat, welches die Anzahl von lateinischen Quadraten der Ordnung log-asymptotisch bestimmt, untersuchen wir die Frage, wie viele -MOLS der Ordnung es gibt. Dies wurde bereits von Donovan und Grannell und Keevash und Luria studiert. Wir verbessern die beste obere Schranke f\"ur einen breiten Bereich von Parametern . Zusätzlich bestimmen wir log-asymptotisch zu wie viele anderen lateinischen Quadraten ein lateinisches Quadrat orthogonal sein kann.
Im dritten Teil studieren wir, wie viele Hyperebenen notwendig sind, um die Punkte eines endlichen Gitters zu überdecken, sodass ein bestimmter Punkt maximal -mal bedeckt ist und alle andere mindestens -mal. Wir untersuchen diese Anzahl f\"ur das Gitter asymptotisch und sogar genau, wenn eins von und viel größer als das andere ist. Dies verallgemeinert ein Ergebnis von Jamison für den Fall . Au{\ss}erdem betrachten wir dieses Problem f\"ur Gitter im reellen Vektorraum, wenn der spezielle Punkt überhaupt nicht bedeckt ist. Dies ist durch die Arbeiten von Clifton und Huang und Sauermann und Wigderson motiviert, die den Hyperwürfel untersucht haben. Wir konzentrieren uns auf zwei-dimensionale Gitter und zeigen, dass schon diese sich sehr unterschiedlich verhalten können
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