23 research outputs found
An asymptotic bound for the strong chromatic number
The strong chromatic number of a graph on
vertices is the least number with the following property: after adding isolated vertices to and taking the union with any
collection of spanning disjoint copies of in the same vertex set, the
resulting graph has a proper vertex-colouring with colours.
We show that for every and every graph on vertices with
, , which is
asymptotically best possible.Comment: Minor correction, accepted for publication in Combin. Probab. Compu
Dependent k-Set Packing on Polynomoids
Specialized hereditary systems, e.g., matroids, are known to have many applications in algorithm design. We define a new notion called d-polynomoid as a hereditary system (E, ? ? 2^E) so that every two maximal sets in ? have less than d elements in common. We study the problem that, given a d-polynomoid (E, ?), asks if the ground set E contains ? disjoint k-subsets that are not in ?, and obtain a complexity trichotomy result for all pairs of k ? 1 and d ? 0. Our algorithmic result yields a sufficient and necessary condition that decides whether each hypergraph in some classes of r-uniform hypergraphs has a perfect matching, which has a number of algorithmic applications
Random groups and Property (T): \.Zuk's theorem revisited
We provide a full and rigorous proof of a theorem attributed to \.Zuk,
stating that random groups in the Gromov density model for d > 1/3 have
property (T) with high probability. The original paper had numerous gaps, in
particular, crucial steps involving passing between different models of random
groups were not described. We fix the gaps using combinatorial arguments and a
recent result concerning perfect matchings in random hypergraphs. We also
provide an alternative proof, avoiding combinatorial difficulties and relying
solely on spectral properties of random graphs in G(n, p) model.Comment: v2: minor correction
Cooperative conditions for the existence of rainbow matchings
Let , and let be a family of non-empty sets of
edges in a bipartite graph. If the union of every members of
contains a matching of size , then there exists an -rainbow
matching of size . Upon replacing by , the result can be
proved both topologically and by a relatively simple combinatorial argument.
The main effort is in gaining the last , which makes the result sharp