50 research outputs found
Perfect 2-colorings of the grassmann graph of planes
We construct an infinite family of intriguing sets, or equivalently perfect 2-colorings, that are not tight in the Grassmann graph of planes of PG(n, q), n ≥ 5 odd, and show that the members of the family are the smallest possible examples if n ≥ 9 or q ≥ 25
Maximal cocliques and the chromatic number of the Kneser graph on chambers of PG
Let be the graph whose vertices are the chambers of the finite
projective -space PG, with two vertices being adjacent if and only if
the corresponding chambers are in general position. We show that a maximal
independent set of vertices of contains , or
, or at most elements. For the
structure of the largest maximal independent sets is described. For
the structure of the maximal independent sets of the three largest
cardinalities is described. Using the cardinality of the second largest maximal
independent sets, we show that the chromatic number of is
Sets of generators blocking all generators in finite classical polar spaces
We introduce generator blocking sets of finite classical polar spaces. These
sets are a generalisation of maximal partial spreads. We prove a
characterization of these minimal sets of the polar spaces Q(2n,q), Q-(2n+1,q)
and H(2n,q^2), in terms of cones with vertex a subspace contained in the polar
space and with base a generator blocking set in a polar space of rank 2.Comment: accepted for J. Comb. Theory
Partial ovoids and partial spreads in finite classical polar spaces
We survey the main results on ovoids and spreads, large maximal partial ovoids and large maximal partial spreads, and on small maximal partial ovoids and small maximal partial spreads in classical finite polar spaces. We also discuss the main results on the spectrum problem on maximal partial ovoids and maximal partial spreads in classical finite polar spaces
On the smallest non-trivial tight sets in Hermitian polar spaces
We show that an x-tight set of the Hermitian polar spaces H(4; q(2)) and H(6; q(2)) respectively, is the union of x disjoint generators of the polar space provided that x is small compared to q. For H(4; q(2)) we need the bound x < q + 1 and we can show that this bound is sharp