174 research outputs found
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
Hamiltonicity, Pancyclicity, and Cycle Extendability in Graphs
The study of cycles, particularly Hamiltonian cycles, is very important in many applications.
Bondy posited his famous metaconjecture, that every condition sufficient for Hamiltonicity actually guarantees a graph is pancyclic. Pancyclicity is a stronger structural property than Hamiltonicity.
An even stronger structural property is for a graph to be cycle extendable. Hendry conjectured that any graph which is Hamiltonian and chordal is cycle extendable.
In this dissertation, cycle extendability is investigated and generalized. It is proved that chordal 2-connected K1,3-free graphs are cycle extendable. S-cycle extendability was defined by Beasley and Brown, where S is any set of positive integers. A conjecture is presented that Hamiltonian chordal graphs are {1, 2}-cycle extendable.
Dirac’s Theorem is an classic result establishing a minimum degree condition for a graph to be Hamiltonian. Ore’s condition is another early result giving a sufficient condition for Hamiltonicity. In this dissertation, generalizations of Dirac’s and Ore’s Theorems are presented.
The Chvatal-Erdos condition is a result showing that if the maximum size of an independent set in a graph G is less than or equal to the minimum number of vertices whose deletion increases the number of components of G, then G is Hamiltonian. It is proved here that the Chvatal-Erdos condition guarantees that a graph is cycle extendable. It is also shown that a graph having a Hamiltonian elimination ordering is cycle extendable.
The existence of Hamiltonian cycles which avoid sets of edges of a certain size and certain subgraphs is a new topic recently investigated by Harlan, et al., which clearly has applications to scheduling and communication networks among other things. The theory is extended here to bipartite graphs. Specifically, the conditions for the existence of a Hamiltonian cycle that avoids edges, or some subgraph of a certain size, are determined for the bipartite case.
Briefly, this dissertation contributes to the state of the art of Hamiltonian cycles, cycle extendability and edge and graph avoiding Hamiltonian cycles, which is an important area of graph theory
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
Friends and Strangers Walking on Graphs
Given graphs and with vertex sets and of the same
cardinality, we define a graph whose vertex set consists of
all bijections , where two bijections and
are adjacent if they agree everywhere except for two adjacent
vertices such that and are adjacent in
. This setup, which has a natural interpretation in terms of friends and
strangers walking on graphs, provides a common generalization of Cayley graphs
of symmetric groups generated by transpositions, the famous -puzzle,
generalizations of the -puzzle as studied by Wilson, and work of Stanley
related to flag -vectors. We derive several general results about the graphs
before focusing our attention on some specific choices of
. When is a path graph, we show that the connected components of
correspond to the acyclic orientations of the complement of
. When is a cycle, we obtain a full description of the connected
components of in terms of toric acyclic orientations of the
complement of . We then derive various necessary and/or sufficient
conditions on the graphs and that guarantee the connectedness of
. Finally, we raise several promising further questions.Comment: 28 pages, 5 figure
Boundary properties of graphs
A set of graphs may acquire various desirable properties, if we apply suitable restrictions
on the set. We investigate the following two questions: How far, exactly, must one restrict
the structure of a graph to obtain a certain interesting property? What kind of tools are
helpful to classify sets of graphs into those which satisfy a property and those that do not?
Equipped with a containment relation, a graph class is a special example of a partially
ordered set. We introduce the notion of a boundary ideal as a generalisation of a notion
introduced by Alekseev in 2003, to provide a tool to indicate whether a partially ordered set
satisfies a desirable property or not. This tool can give a complete characterisation of lower
ideals defined by a finite forbidden set, into those that satisfy the given property and to
those that do not. In the case of graphs, a lower ideal with respect to the induced subgraph
relation is known as a hereditary graph class.
We study three interrelated types of properties for hereditary graph classes: the existence
of an efficient solution to an algorithmic graph problem, the boundedness of the graph
parameter known as clique-width, and well-quasi-orderability by the induced subgraph relation.
It was shown by Courcelle, Makowsky and Rotics in 2000 that, for a graph class, boundedness
of clique-width immediately implies an efficient solution to a wide range of algorithmic
problems. This serves as one of the motivations to study clique-width. As for well-quasiorderability,
we conjecture that every hereditary graph class that is well-quasi-ordered by
the induced subgraph relation also has bounded clique-width.
We discover the first boundary classes for several algorithmic graph problems, including
the Hamiltonian cycle problem. We also give polynomial-time algorithms for the dominating
induced matching problem, for some restricted graph classes.
After discussing the special importance of bipartite graphs in the study of clique-width,
we describe a general framework for constructing bipartite graphs of large clique-width. As
a consequence, we find a new minimal class of unbounded clique-width.
We prove numerous positive and negative results regarding the well-quasi-orderability of
classes of bipartite graphs. This completes a characterisation of the well-quasi-orderability of
all classes of bipartite graphs defined by one forbidden induced bipartite subgraph. We also
make considerable progress in characterising general graph classes defined by two forbidden
induced subgraphs, reducing the task to a small finite number of open cases. Finally, we
show that, in general, for hereditary graph classes defined by a forbidden set of bounded
finite size, a similar reduction is not usually possible, but the number of boundary classes
to determine well-quasi-orderability is nevertheless finite.
Our results, together with the notion of boundary ideals, are also relevant for the study
of other partially ordered sets in mathematics, such as permutations ordered by the pattern
containment relation
The k-fixed-endpoint path partition problem
The Hamiltonian path problem is to determine whether a graph has a Hamiltonian path. This problem is NP-complete in general. The path partition problem is to determine the minimum number of vertex-disjoint paths required to cover a graph. Since this problem is a generalization of the Hamiltonian path problem, it is also NP-complete in general. The k-fixed-endpoint path partition problem is to determine the minimum number of vertex-disjoint paths required to cover a graphG such that each vertex in a set T of k vertices is an endpoint of a path. Since this problem is a generalization of the Hamiltonian path problem and path partition problem, it is also NP-complete in general. For certain classes of graphs, there exist efficient algorithms for the k-fixed-endpoint path partition problem. We consider this problem restricted to trees, threshold graphs, block graphs, and unit interval graphs and show min-max theorems which characterize the k-fixed-endpoint pathpartition number
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