187 research outputs found
Spanning trees in random graphs
For each , we prove that there exists some for which
the binomial random graph almost surely contains a copy of
every tree with vertices and maximum degree at most . In doing so,
we confirm a conjecture by Kahn.Comment: 71 pages, 31 figures, version accepted for publication in Advances in
Mathematic
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
Generation and New Infinite Families of -hypohamiltonian Graphs
We present an algorithm which can generate all pairwise non-isomorphic
-hypohamiltonian graphs, i.e. non-hamiltonian graphs in which the removal
of any pair of adjacent vertices yields a hamiltonian graph, of a given order.
We introduce new bounding criteria specifically designed for
-hypohamiltonian graphs, allowing us to improve upon earlier computational
results. Specifically, we characterise the orders for which
-hypohamiltonian graphs exist and improve existing lower bounds on the
orders of the smallest planar and the smallest bipartite -hypohamiltonian
graphs. Furthermore, we describe a new operation for creating
-hypohamiltonian graphs that preserves planarity under certain conditions
and use it to prove the existence of a planar -hypohamiltonian graph of
order for every integer . Additionally, motivated by a theorem
of Thomassen on hypohamiltonian graphs, we show the existence
-hypohamiltonian graphs with large maximum degree and size.Comment: 21 page
B-VPG Representation of AT-free Outerplanar Graphs
B-VPG graphs are intersection graphs of axis-parallel line segments in
the plane. In this paper, we show that all AT-free outerplanar graphs are
B-VPG. We first prove that every AT-free outerplanar graph is an induced
subgraph of a biconnected outerpath (biconnected outerplanar graphs whose weak
dual is a path) and then we design a B-VPG drawing procedure for
biconnected outerpaths. Our proofs are constructive and give a polynomial time
B-VPG drawing algorithm for the class.
We also characterize all subgraphs of biconnected outerpaths and name this
graph class "linear outerplanar". This class is a proper superclass of AT-free
outerplanar graphs and a proper subclass of outerplanar graphs with pathwidth
at most 2. It turns out that every graph in this class can be realized both as
an induced subgraph and as a spanning subgraph of (different) biconnected
outerpaths.Comment: A preliminary version, which did not contain the characterization of
linear outerplanar graphs (Section 3), was presented in the
International Conference on Algorithms and Discrete Applied Mathematics
(CALDAM) 2022. The definition of linear outerplanar graphs in this paper
differs from that in the preliminary version and hence Section 4 is ne
Sequential importance sampling for estimating expectations over the space of perfect matchings
This paper makes three contributions to estimating the number of perfect
matching in bipartite graphs. First, we prove that the popular sequential
importance sampling algorithm works in polynomial time for dense bipartite
graphs. More carefully, our algorithm gives a -approximation for
the number of perfect matchings of a -dense bipartite graph, using
samples. With size on
each side and for , a -dense bipartite graph
has all degrees greater than .
Second, practical applications of the algorithm requires many calls to
matching algorithms. A novel preprocessing step is provided which makes
significant improvements.
Third, three applications are provided. The first is for counting Latin
squares, the second is a practical way of computing the greedy algorithm for a
card guessing game with feedback, and the third is for stochastic block models.
In all three examples, sequential importance sampling allows treating practical
problems of reasonably large sizes
The bandwidth theorem for locally dense graphs
The Bandwidth theorem of B\"ottcher, Schacht and Taraz gives a condition on
the minimum degree of an -vertex graph that ensures contains every
-chromatic graph on vertices of bounded degree and of bandwidth
, thereby proving a conjecture of Bollob\'as and Koml\'os. In this paper
we prove a version of the Bandwidth theorem for locally dense graphs. Indeed,
we prove that every locally dense -vertex graph with contains as a subgraph any given (spanning) with bounded
maximum degree and sublinear bandwidth.Comment: 35 pages. Author accepted version, to appear in Forum of Mathematics,
Sigm
Killing a Vortex
The Graph Minors Structure Theorem of Robertson and Seymour asserts that, for
every graph every -minor-free graph can be obtained by clique-sums of
``almost embeddable'' graphs. Here a graph is ``almost embeddable'' if it can
be obtained from a graph of bounded Euler-genus by pasting graphs of bounded
pathwidth in an ``orderly fashion'' into a bounded number of faces, called the
\textit{vortices}, and then adding a bounded number of additional vertices,
called \textit{apices}, with arbitrary neighborhoods. Our main result is a
{full classification} of all graphs for which the use of vortices in the
theorem above can be avoided. To this end we identify a (parametric) graph
and prove that all -minor-free graphs can be
obtained by clique-sums of graphs embeddable in a surface of bounded
Euler-genus after deleting a bounded number of vertices. We show that this
result is tight in the sense that the appearance of vortices cannot be avoided
for -minor-free graphs, whenever is not a minor of for
some
Using our new structure theorem, we design an algorithm that, given an
-minor-free graph computes the generating function of all
perfect matchings of in polynomial time. Our results, combined with known
complexity results, imply a complete characterization of minor-closed graph
classes where the number of perfect matchings is polynomially computable: They
are exactly those graph classes that do not contain every as
a minor. This provides a \textit{sharp} complexity dichotomy for the problem of
counting perfect matchings in minor-closed classes.Comment: An earlier version of this paper has appeared at FOCS 2022 We also
changed the term "vga-hierarchy" with the more appropriate term
"vga-lattice". arXiv admin note: text overlap with arXiv:2010.12397 by other
author
Characterizing Forbidden Pairs for Hamiltonian Properties
https://digitalcommons.memphis.edu/speccoll-faudreerj/1207/thumbnail.jp
- …