33,123 research outputs found

    Tverberg's theorem and graph coloring

    Full text link
    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. Updated languag

    On the intersection of distance-jj-ovoids and subpolygons in generalized polygons

    Get PDF
    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 (s′,t′)(s',t') of a generalized hexagon of order intersectinaconstantnumberofpoints.Iftheautomorphismgroupofthegeneralizedhexagonissufficientlylarge,thenthisisthecaseifandonlyif=s′t′ intersect in a constant number of points. If the automorphism group of the generalized hexagon is sufficiently large, then this is the case if and only if =s't'. We derive a similar result for the intersection of distance-2-ovoids and suboctagons of generalized octagons

    Restricted rr-Stirling Numbers and their Combinatorial Applications

    Get PDF
    We study set partitions with rr distinguished elements and block sizes found in an arbitrary index set SS. The enumeration of these (S,r)(S,r)-partitions leads to the introduction of (S,r)(S,r)-Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the rr-Stirling numbers. We also introduce the associated (S,r)(S,r)-Bell and (S,r)(S,r)-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For SS with some extra structure, we show that the inverse of the (S,r)(S,r)-Stirling matrix encodes the M\"obius functions of two families of posets. Through several examples, we demonstrate that for some SS the matrices and their inverses involve the enumeration sequences of several combinatorial objects. Further, we highlight how the (S,r)(S,r)-Stirling numbers naturally arise in the enumeration of cliques and acyclic orientations of special graphs, underlining their ubiquity and importance. Finally, we introduce related (S,r)(S,r) generalizations of the poly-Bernoulli and poly-Cauchy numbers, uniting many past works on generalized combinatorial sequences
    • …
    corecore