32 research outputs found
Colouring Complete Multipartite and Kneser-type Digraphs
The dichromatic number of a digraph is the smallest such that can
be partitioned into acyclic subdigraphs, and the dichromatic number of an
undirected graph is the maximum dichromatic number over all its orientations.
Extending a well-known result of Lov\'{a}sz, we show that the dichromatic
number of the Kneser graph is and that the
dichromatic number of the Borsuk graph is if is large
enough. We then study the list version of the dichromatic number. We show that,
for any and , the list
dichromatic number of is . This extends a recent
result of Bulankina and Kupavskii on the list chromatic number of ,
where the same behaviour was observed. We also show that for any ,
and , the list dichromatic number of the complete
-partite graph with vertices in each part is , extending
a classical result of Alon. Finally, we give a directed analogue of Sabidussi's
theorem on the chromatic number of graph products.Comment: 15 page
On the central levels problem
The \emph{central levels problem} asserts that the subgraph of the -dimensional hypercube induced by all bitstrings with at least many 1s and at most many 1s, i.e., the vertices in the middle levels, has a Hamilton cycle for any and .
This problem was raised independently by Buck and Wiedemann, Savage, Gregor and {\v{S}}krekovski, and by Shen and Williams, and it is a common generalization of the well-known \emph{middle levels problem}, namely the case , and classical binary Gray codes, namely the case .
In this paper we present a general constructive solution of the central levels problem.
Our results also imply the existence of optimal cycles through any sequence of consecutive levels in the -dimensional hypercube for any and .
Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the -dimensional hypercube, , that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code
Dimension and Ramsey results in partially ordered sets.
In this dissertation, there are two major parts. One is the dimension results on different classes of partially ordered sets. We developed new tools and theorems to solve the bounds on interval orders using different number of lengths. We also discussed the dimension of interval orders that have a representation with interval lengths in a certain range. We further discussed the interval dimension and semi dimension for posets. In the second part, we discussed several related results on the Ramsey theory of grids, the results involve the application of Product Ramsey Theorem and Partition Ramsey Theore
A collection of open problems in celebration of Imre Leader's 60th birthday
One of the great pleasures of working with Imre Leader is to experience his
infectious delight on encountering a compelling combinatorial problem. This
collection of open problems in combinatorics has been put together by a subset
of his former PhD students and students-of-students for the occasion of his
60th birthday. All of the contributors have been influenced (directly or
indirectly) by Imre: his personality, enthusiasm and his approach to
mathematics. The problems included cover many of the areas of combinatorial
mathematics that Imre is most associated with: including extremal problems on
graphs, set systems and permutations, and Ramsey theory. This is a personal
selection of problems which we find intriguing and deserving of being better
known. It is not intended to be systematic, or to consist of the most
significant or difficult questions in any area. Rather, our main aim is to
celebrate Imre and his mathematics and to hope that these problems will make
him smile. We also hope this collection will be a useful resource for
researchers in combinatorics and will stimulate some enjoyable collaborations
and beautiful mathematics
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
Understanding and Enhancing CDCL-based SAT Solvers
Modern conflict-driven clause-learning (CDCL) Boolean satisfiability (SAT) solvers routinely
solve formulas from industrial domains with millions of variables and clauses, despite the Boolean
satisfiability problem being NP-complete and widely regarded as intractable in general. At the
same time, very small crafted or randomly generated formulas are often infeasible for CDCL
solvers. A commonly proposed explanation is that these solvers somehow exploit the underlying
structure inherent in industrial instances. A better understanding of the structure of Boolean
formulas not only enables improvements to modern SAT solvers, but also lends insight as to why
solvers perform well or poorly on certain types of instances. Even further, examining solvers
through the lens of these underlying structures can help to distinguish the behavior of different
solving heuristics, both in theory and practice.
The first issue we address relates to the representation of SAT formulas. A given Boolean
satisfiability problem can be represented in arbitrarily many ways, and the type of encoding can
have significant effects on SAT solver performance. Further, in some cases, a direct encoding
to SAT may not be the best choice. We introduce a new system that integrates SAT solving
with computer algebra systems (CAS) to address representation issues for several graph-theoretic
problems. We use this system to improve the bounds on several finitely-verified conjectures
related to graph-theoretic problems. We demonstrate how our approach is more appropriate for
these problems than other off-the-shelf SAT-based tools.
For more typical SAT formulas, a better understanding of their underlying structural properties,
and how they relate to SAT solving, can deepen our understanding of SAT. We perform a largescale
evaluation of many of the popular structural measures of formulas, such as community
structure, treewidth, and backdoors. We investigate how these parameters correlate with CDCL
solving time, and whether they can effectively be used to distinguish formulas from different
domains. We demonstrate how these measures can be used as a means to understand the behavior
of solvers during search. A common theme is that the solver exhibits locality during search
through the lens of these underlying structures, and that the choice of solving heuristic can greatly
influence this locality. We posit that this local behavior of modern SAT solvers is crucial to their
performance.
The remaining contributions dive deeper into two new measures of SAT formulas. We first
consider a simple measure, denoted “mergeability,” which characterizes the proportion of input
clauses pairs that can resolve and merge. We develop a formula generator that takes as input a seed
formula, and creates a sequence of increasingly more mergeable formulas, while maintaining many
of the properties of the original formula. Experiments over randomly-generated industrial-like
instances suggest that mergeability strongly negatively correlates with CDCL solving time, i.e., as
the mergeability of formulas increases, the solving time decreases, particularly for unsatisfiable
instances.
Our final contribution considers whether one of the aforementioned measures, namely backdoor
size, is influenced by solver heuristics in theory. Starting from the notion of learning-sensitive
(LS) backdoors, we consider various extensions of LS backdoors by incorporating different branching
heuristics and restart policies. We introduce learning-sensitive with restarts (LSR) backdoors
and show that, when backjumping is disallowed, LSR backdoors may be exponentially smaller
than LS backdoors. We further demonstrate that the size of LSR backdoors are dependent on the
learning scheme used during search. Finally, we present new algorithms to compute upper-bounds
on LSR backdoors that intrinsically rely upon restarts, and can be computed with a single run of
a SAT solver. We empirically demonstrate that this can often produce smaller backdoors than
previous approaches to computing LS backdoors