72 research outputs found
Almost partitioning the hypercube into copies of a graph
Let H be an induced subgraph of the hypercube Qk, for some k. We show that for some c=c(H), the vertices of Qn can be partitioned into induced copies of H and a remainder of at most O(nc) vertices. We also show that the error term cannot be replaced by anything smaller than log
Almost partitioning the hypercube into copies of a graph
Let H be an induced subgraph of the hypercube Qk, for some k. We show that for some c=c(H), the vertices of Qn can be partitioned into induced copies of H and a remainder of at most O(nc) vertices. We also show that the error term cannot be replaced by anything smaller than log
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Tilings and other combinatorial results
In this dissertation we treat three tiling problems and three problems in combinatorial geometry, extremal graph theory and sparse Ramsey theory.
We first consider tilings of . In this setting a tile is just a finite subset of . We say that tiles if the latter set admits a partition into isometric copies of . Chalcraft observed that there exist that do not tile but tile for some . He conjectured that such exists for any given tile. We prove this conjecture in Chapter 2.
In Chapter 3 we prove a conjecture of Lonc, stating that for any poset of size a power of , if has a greatest and a least element, then there is a positive integer such that can be partitioned into copies of .
The third tiling problem is about vertex-partitions of the hypercube graph . Offner asked: if is a subgraph of such is a power of , must , for some , admit a partition into isomorphic copies of ? In Chapter 4 we answer this question in the affirmative.
We follow up with a question in combinatorial geometry. A line in a planar set is a maximal collinear subset of . P\'or and Wood considered colourings of finite without large lines with a bounded number of colours. In particular, they examined whether monochromatic lines always appear in such colourings provided that is large. They conjectured that for all there exists an such that if and does not contain a line of cardinality larger than , then every colouring of with colours produces a monochromatic line. In Chapter 5 we construct arbitrarily large counterexamples for the case .
We follow up with a problem in extremal graph theory. For any graph, we say that a given edge is triangular if it forms a triangle with two other edges. How few triangular edges can there be in a graph with vertices and edges? For sufficiently large we prove a conjecture of F\"uredi and Maleki that gives an exact formula for this minimum. This proof is given in Chapter 6.
Finally, Chapter 7 is concerned with degrees of vertices in directed hypergraphs. One way to prescribe an orientation to an -uniform graph is to assign for each of its edges one of the possible orderings of its elements. Then, for any -set of vertices and any -set of indices , we define the -degree of to be the number of edges containing vertices in precisely the positions labelled by . Caro and Hansberg were interested in determining whether a given -uniform hypergraph admits an orientation where every set of vertices has some -degree equal to . They conjectured that a certain Hall-type condition is sufficient. We show that this is true for large, but false in general.EPSR
FICS 2010
International audienceInformal proceedings of the 7th workshop on Fixed Points in Computer Science (FICS 2010), held in Brno, 21-22 August 201
Computational complexity of counting coincidences
Can you decide if there is a coincidence in the numbers counting two
different combinatorial objects? For example, can you decide if two regions in
have the same number of domino tilings? There are two versions
of the problem, with and boxes. We
prove that in both cases the coincidence problem is not in the polynomial
hierarchy unless the polynomial hierarchy collapses to a finite level. While
the conclusions are the same, the proofs are notably different and generalize
in different directions.
We proceed to explore the coincidence problem for counting independent sets
and matchings in graphs, matroid bases, order ideals and linear extensions in
posets, permutation patterns, and the Kronecker coefficients. We also make a
number of conjectures for counting other combinatorial objects such as plane
triangulations, contingency tables, standard Young tableaux, reduced
factorizations and the Littlewood--Richardson coefficients.Comment: 23 pages, 6 figure
Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras
We study a non-commutative generalization of Stone duality that connects a
class of inverse semigroups, called Boolean inverse -semigroups, with a
class of topological groupoids, called Hausdorff Boolean groupoids. Much of the
paper is given over to showing that Boolean inverse -semigroups arise
as completions of inverse semigroups we call pre-Boolean. An inverse
-semigroup is pre-Boolean if and only if every tight filter is an
ultrafilter, where the definition of a tight filter is obtained by combining
work of both Exel and Lenz. A simple necessary condition for a semigroup to be
pre-Boolean is derived and a variety of examples of inverse semigroups are
shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees
matrix semigroups over the polycyclics, are pre-Boolean and it is proved that
the groups of units of their completions are precisely the Thompson-Higman
groups . The inverse semigroups arising from suitable directed graphs
are also pre-Boolean and the topological groupoids arising from these graph
inverse semigroups under our non-commutative Stone duality are the groupoids
that arise from the Cuntz-Krieger -algebras.Comment: The presentation has been sharpened up and some minor errors
correcte
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