6,836 research outputs found
Cuts in matchings of 3-connected cubic graphs
We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette,
Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and
on even graphs in digraphs whose contraction is strongly connected
(Hochst\"attler). We show that all of them fit into the same framework related
to cuts in matchings. This allows us to find a counterexample to the conjecture
of Hochst\"attler and show that the conjecture of Neumann-Lara holds for all
planar graphs on at most 26 vertices. Finally, we state a new conjecture on
bipartite cubic oriented graphs, that naturally arises in this setting.Comment: 12 pages, 5 figures, 1 table. Improved expositio
Some snarks are worse than others
Many conjectures and open problems in graph theory can either be reduced to
cubic graphs or are directly stated for cubic graphs. Furthermore, it is known
that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless
cubic graph which is not 3--edge-colourable. In this paper we deal with the
fact that the family of potential counterexamples to many interesting
conjectures can be narrowed even further to the family of
bridgeless cubic graphs whose edge set cannot be covered with four perfect
matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover
Conjecture and the Fan-Raspaud Conjecture are examples of statements for which
is crucial. In this paper, we study parameters which have
the potential to further refine and thus enlarge the set of
cubic graphs for which the mentioned conjectures can be verified. We show that
can be naturally decomposed into subsets with increasing
complexity, thereby producing a natural scale for proving these conjectures.
More precisely, we consider the following parameters and questions: given a
bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii)
how many copies of the same perfect matching need to be added, and (iii) how
many 2--factors need to be added so that the resulting regular graph is Class
I? We present new results for these parameters and we also establish some
strong relations between these problems and some long-standing conjectures.Comment: 27 pages, 16 figure
Decompositions into spanning rainbow structures
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares and has been the focus of extensive research ever since. Euler posed a problem equivalent to finding properly n-edge-coloured complete bipartite graphs Kn,n which can be decomposed into rainbow perfect matchings. While there are proper edge-colourings of Kn,n without even a single rainbow perfect matching, the theme of this paper is to show that with some very weak additional constraints one can find many disjoint rainbow perfect matchings. In particular, we prove that if some fraction of the colour classes have at most (1−o(1))n edges then one can nearly-decompose the edges of Kn,n into edge-disjoint perfect rainbow matchings. As an application of this, we establish in a very strong form a conjecture of Akbari and Alipour and asymptotically prove a conjecture of Barat and Nagy. Both these conjectures concern rainbow perfect matchings in edge-colourings of Kn,n with quadratically many colours.
Using our techniques, we also prove a number of results on near-decompositions of graphs into other rainbow structures like Hamiltonian cycles and spanning trees. Most notably, we prove that any properly coloured complete graph can be nearly-decomposed into spanning rainbow trees. This asymptotically proves the Brualdi-Hollingsworth and Kaneko-Kano-Suzuki conjectures which predict that a perfect decomposition should exist under the same assumptions
Equistarable graphs and counterexamples to three conjectures on equistable graphs
Equistable graphs are graphs admitting positive weights on vertices such that
a subset of vertices is a maximal stable set if and only if it is of total
weight . In , Mahadev et al.~introduced a subclass of equistable
graphs, called strongly equistable graphs, as graphs such that for every and every non-empty subset of vertices that is not a maximal stable set,
there exist positive vertex weights such that every maximal stable set is of
total weight and the total weight of does not equal . Mahadev et al.
conjectured that every equistable graph is strongly equistable. General
partition graphs are the intersection graphs of set systems over a finite
ground set such that every maximal stable set of the graph corresponds to a
partition of . In , Orlin proved that every general partition graph is
equistable, and conjectured that the converse holds as well.
Orlin's conjecture, if true, would imply the conjecture due to Mahadev,
Peled, and Sun. An intermediate conjecture, one that would follow from Orlin's
conjecture and would imply the conjecture by Mahadev, Peled, and Sun, was posed
by Miklavi\v{c} and Milani\v{c} in , and states that every equistable
graph has a clique intersecting all maximal stable sets. The above conjectures
have been verified for several graph classes. We introduce the notion of
equistarable graphs and based on it construct counterexamples to all three
conjectures within the class of complements of line graphs of triangle-free
graphs
Conditional Lower Bounds for Space/Time Tradeoffs
In recent years much effort has been concentrated towards achieving
polynomial time lower bounds on algorithms for solving various well-known
problems. A useful technique for showing such lower bounds is to prove them
conditionally based on well-studied hardness assumptions such as 3SUM, APSP,
SETH, etc. This line of research helps to obtain a better understanding of the
complexity inside P.
A related question asks to prove conditional space lower bounds on data
structures that are constructed to solve certain algorithmic tasks after an
initial preprocessing stage. This question received little attention in
previous research even though it has potential strong impact.
In this paper we address this question and show that surprisingly many of the
well-studied hard problems that are known to have conditional polynomial time
lower bounds are also hard when concerning space. This hardness is shown as a
tradeoff between the space consumed by the data structure and the time needed
to answer queries. The tradeoff may be either smooth or admit one or more
singularity points.
We reveal interesting connections between different space hardness
conjectures and present matching upper bounds. We also apply these hardness
conjectures to both static and dynamic problems and prove their conditional
space hardness.
We believe that this novel framework of polynomial space conjectures can play
an important role in expressing polynomial space lower bounds of many important
algorithmic problems. Moreover, it seems that it can also help in achieving a
better understanding of the hardness of their corresponding problems in terms
of time
Kasteleyn cokernels
We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in
enumerating matchings of planar graphs, up to matrix operations on their rows
and columns. If such a matrix is defined over a principal ideal domain, this is
equivalent to considering its Smith normal form or its cokernel. Many
variations of the enumeration methods result in equivalent matrices. In
particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus
matrices.
We apply these ideas to plane partitions and related planar of tilings. We
list a number of conjectures, supported by experiments in Maple, about the
forms of matrices associated to enumerations of plane partitions and other
lozenge tilings of planar regions and their symmetry classes. We focus on the
case where the enumerations are round or -round, and we conjecture that
cokernels remain round or -round for related ``impossible enumerations'' in
which there are no tilings. Our conjectures provide a new view of the topic of
enumerating symmetry classes of plane partitions and their generalizations. In
particular we conjecture that a -specialization of a Jacobi-Trudi matrix has
a Smith normal form. If so it could be an interesting structure associated to
the corresponding irreducible representation of \SL(n,\C). Finally we find,
with proof, the normal form of the matrix that appears in the enumeration of
domino tilings of an Aztec diamond.Comment: 14 pages, 19 in-line figures. Very minor copy correction
- …