The second minimum size of a finite subspace partition

Let be a fixed prime power and let denote a vector space of dimension over the Galois field with elements. A subspace partition (also called “vector space partition”) of is a collection of subspaces of with the property that every nonzero element of appears in exactly one of these subspaces. Given positive integers such that , we say a subspace partition of has type if it is composed of subspaces of dimension and subspaces of dimension . Let . In this paper, we prove that if divides , then one can (algebraically) construct every possible subspace partition of of type whenever , where and . Our construction allows us to sequentially reconfigure batches of subspaces of dimension into batches of subspaces of dimension . In particular, this accounts for all numerically allowed subspace partition types of under some additional conditions, for example, when .

A subspace partition Π of V = V(n, q) is a collection of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of Π. The size of Π is the number of its subspaces. Let σq(n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let ρq(n, t) denote the maximum size of a subspace partition of V in which the smallest subspace has dimension t. In this article, we determine the values of σq(n, t) and ρq(n, t) for all positive integers n and t. Furthermore, we prove that if n ≥ 2t, then the minimum size of a maximal partial t-spread in V(n + t -1, q) is σ q(n, t).

The complete characterization of the minimum size supertail in a subspace partition

A vector space partition is here a collection [Formula: see text] of subspaces of a finite vector space V(n, q), of dimension n over a finite field with q elements, with the property that every non-zero vector is contained in a unique member of [Formula: see text]. Vector space partitions relate to finite projective planes, design theory and error correcting codes. In the first part of the paper I will discuss some relations between vector space partitions and other branches of mathematics. The other part of the paper contains a survey of known results on the type of a vector space partition, more precisely: the theorem of Beutelspacher and Heden on T-partitions, rather recent results of El-Zanati et al. on the different types that appear in the spaces V(n, 2), for n ≤ 8, a result of Heden and Lehmann on vector space partitions and maximal partial spreads including their new necessary condition for the existence of a vector space partition, and furthermore, I will give a theorem of Heden on the length of the tail of a vector space partition. Finally, I will also give a few historical remarks.

( s, k, λ)-Partitions of a vector space

Let V = V(n, q) be a vector space of dimension n over the finite field with q elements, and let d 1 < d 2 < ... < d m be the dimensions that occur in a subspace partition $${\mathcal{P}}$$ of V. Let ? q (n, t) denote the minimum size of a subspace partition $${\mathcal P}$$ of V, in which t is the largest dimension of a subspace. For any integer s, with 1 < s ≤ m, the set of subspaces in $${\mathcal{P}}$$ of dimension less than d s is called the s-supertail of $${\mathcal{P}}$$ . The main result is that the number of spaces in an s-supertail is at least ? q (d s , d s?1).

Subspace partitions of [formula omitted] containing direct sums

The minimum size of a finite subspace partition

Let $$V=V(n,q)$$V=V(n,q) denote the vector space of dimension n over the finite field with q elements. A subspace partition$$\mathcal {P}$$P of V is a collection of nontrivial subspaces of V such that each nonzero vector of V is in exactly one subspace of $$\mathcal {P}$$P. For any integer d, the d-supertail of $$\mathcal {P}$$P is the set of subspaces in $$\mathcal {P}$$P of dimension less than d, and it is denoted by ST. Let $$\sigma _q(n,t)$$źq(n,t) denote the minimum number of subspaces in any subspace partition of V in which the largest subspace has dimension t. It was shown by Heden et al. that $$|ST|\ge \sigma _q(d,t)$$|ST|źźq(d,t), where t is the largest dimension of a subspace in ST. In this paper, we show that if $$|ST|=\sigma _q(d,t)$$|ST|=źq(d,t), then the union of all the subspaces in ST constitutes a subspace under certain conditions.

Subspace partitions of [formula omitted] containing direct sums II: General case

Partitions of finite vector spaces over [formula omitted] into subspaces of dimensions 2 and s

The lattice of finite subspace partitions

On the type(s) of minimum size subspace partitions

Let V n (q) denote a vector space of dimension n over the field with q elements. A set $${\mathcal{P}}$$ of subspaces of V n (q) is a partition of V n (q) if every nonzero vector in V n (q) is contained in exactly one subspace in $${\mathcal{P}}$$ . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of V n (q) containing a i subspaces of dimension n i for 1 ? i ? k induces a uniformly resolvable design on q n points with a i parallel classes with block size $$q^{n_i}$$ , 1 ? i ? k, and also corresponds to a factorization of the complete graph $$K_{q^n}$$ into $$a_i K_{q^{n_i}}$$ -factors, 1 ? i ? k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on q n points where corresponding partitions of V n (q) do not exist.

Given a graph G, a q-open neighborhood conflict-free coloring or q-ONCF-coloring is a vertex coloring \(c:V(G) \rightarrow \{1,2,\ldots ,q\}\) such that for each vertex \(v \in V(G)\) there is a vertex in N(v) that is uniquely colored from the rest of the vertices in N(v). When we replace N(v) by the closed neighborhood N[v], then we call such a coloring a q-closed neighborhood conflict-free coloring or simply q-CNCF-coloring. In this paper, we study the NP-hard decision questions of whether for a constant q an input graph has a q-ONCF-coloring or a q-CNCF-coloring. We will study these two problems in the parameterized setting. First of all, we study running time bounds on FPT-algorithms for these problems, when parameterized by treewidth. We improve the existing upper bounds, and also provide lower bounds on the running time under ETH and SETH. Secondly, we study the kernelization complexity of both problems, using vertex cover as the parameter. We show that both \((q \ge 2)\)-ONCF-coloring and \((q \ge 3)\)-CNCF-coloring cannot have polynomial kernels when parameterized by the size of a vertex cover unless \(\mathsf {NP \subseteq coNP/poly}\). On the other hand, we obtain a polynomial kernel for 2-CNCF-coloring parameterized by vertex cover. We conclude the study with some combinatorial results. Denote \(\chi _{ON}(G)\) and \(\chi _{CN}(G)\) to be the minimum number of colors required to ONCF-color and CNCF-color G, respectively. Upper bounds on \(\chi _{CN}(G)\) with respect to structural parameters like minimum vertex cover size, minimum feedback vertex set size and treewidth are known. To the best of our knowledge only an upper bound on \(\chi _{ON}(G)\) with respect to minimum vertex cover size was known. We provide tight bounds for \(\chi _{ON}(G)\) with respect to minimum vertex cover size. Also, we provide the first upper bounds on \(\chi _{ON}(G)\) with respect to minimum feedback vertex set size and treewidth.