34,103 research outputs found
Tverberg's theorem and graph coloring
The topological Tverberg theorem has been generalized in several directions
by setting extra restrictions on the Tverberg partitions.
Restricted Tverberg partitions, defined by the idea that certain points
cannot be in the same part, are encoded with graphs. When two points are
adjacent in the graph, they are not in the same part. If the restrictions are
too harsh, then the topological Tverberg theorem fails. The colored Tverberg
theorem corresponds to graphs constructed as disjoint unions of small complete
graphs. Hell studied the case of paths and cycles.
In graph theory these partitions are usually viewed as graph colorings. As
explored by Aharoni, Haxell, Meshulam and others there are fundamental
connections between several notions of graph colorings and topological
combinatorics.
For ordinary graph colorings it is enough to require that the number of
colors q satisfy q>Delta, where Delta is the maximal degree of the graph. It
was proven by the first author using equivariant topology that if q>\Delta^2
then the topological Tverberg theorem still works. It is conjectured that
q>K\Delta is also enough for some constant K, and in this paper we prove a
fixed-parameter version of that conjecture.
The required topological connectivity results are proven with shellability,
which also strengthens some previous partial results where the topological
connectivity was proven with the nerve lemma.Comment: To appear in Discrete and Computational Geometry, 13 pages, 1 figure.
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On the intersection of distance--ovoids and subpolygons in generalized polygons
De Wispelaere and Van Maldeghem gave a technique for calculating the intersection sizes of combinatorial substructures associated with regular partitions of distance-regular graphs. This technique was based on the orthogonality of the eigenvectors which correspond to distinct eigenvalues of the (symmetric) adjacency matrix. In the present paper, we give a more general method for calculating intersection sizes of combinatorial structures. The proof of this method is based on the solution of a linear system of equations which is obtained by means of double countings. We also give a new class of regular partitions of generalized hexagons and determine under which conditions ovoids and subhexagons of order of a generalized hexagon of order . We derive a similar result for the intersection of distance-2-ovoids and suboctagons of generalized octagons
Restricted -Stirling Numbers and their Combinatorial Applications
We study set partitions with distinguished elements and block sizes found
in an arbitrary index set . The enumeration of these -partitions
leads to the introduction of -Stirling numbers, an extremely
wide-ranging generalization of the classical Stirling numbers and the
-Stirling numbers. We also introduce the associated -Bell and
-factorial numbers. We study fundamental aspects of these numbers,
including recurrence relations and determinantal expressions. For with some
extra structure, we show that the inverse of the -Stirling matrix
encodes the M\"obius functions of two families of posets. Through several
examples, we demonstrate that for some the matrices and their inverses
involve the enumeration sequences of several combinatorial objects. Further, we
highlight how the -Stirling numbers naturally arise in the enumeration
of cliques and acyclic orientations of special graphs, underlining their
ubiquity and importance. Finally, we introduce related generalizations
of the poly-Bernoulli and poly-Cauchy numbers, uniting many past works on
generalized combinatorial sequences
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