We consider two general frameworks for multiple domination, which are called 〈r, s〉-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the 〈r, s〉-domination and parametric domination numbers. These generalisations are based on t...

We show that whenever a contractive k-tuple T on a finite dimensional space H has a unitary dilation, then for any fixed degree N there is a unitary k-tuple U on a finite dimensional space so that q(T ) = PHq(U)|H for all polynomials q of degree at most N .

We show that the qualitative form of the prime k-tuple conjecture implies a certain conjecture related to the least size of the interval containing exactly ` prime numbers. Also, we prove that the latter conjecture implies the prime k-tuple conjecture for some minimal admissible set. The proof is self-contained and elementary.

The three domatic number problem asks whether a given undirected graph can be partitioned into at least three dominating sets, i.e., sets whose closed neighborhood equals the vertex set of the graph. Since this problem is NP-complete, no polynomial-time algorithm is known for it. The naive deterministic algorithm for this problem runs in time 3n, up to polynomial factors. In this paper, we desi...

Let Ω be an irreducible bounded symmetric domain of rank r in ℂd. $$\mathbb{K}$$ the maximal compact subgroup identity component G biholomorphic automorphism group Ω. The consisting linear transformations acts naturally on any d-tuple T = (T1, …, Td) commuting operators. If orbit this action modulo unitary equivalence is a singleton, then we say that -homogeneous. In paper, obtain model for cer...

For a connected graph G of order |V (G)| ≥ 3 and a k-labelling c : E(G) → {1, 2, . . . , k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (l1, l2, . . . , lk), where li is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detect...

We prove a new structural property regarding the “skyline” of uniform radius disks and use this to derive a number of new sequential and distributed approximation algorithms for well-known optimization problems on unit disk graphs (UDGs). Specifically, the paper presents new approximation algorithms for two problems: domatic partition and weighted minimum dominating set (WMDS) on UDGs, both of ...

A transversal of a hypergraph is a set of vertices meeting all the hyperedges. A k-fold transversal of a hypergraph is a set of vertices such that every hyperedge has at least k elements of. In this paper, we prove that a k-fold transversal of a balanced hypergraph can be expressed as a union of k pairwise disjoint transversals and such partition can be obtained in polynomial time. We give an N...

Consider a set of N objects, consisting of Ni objects originating from population Ai, i = 1,. . . , k, so that N = Zf-, Ni. In order to classify the set of N objects, we measure a variable with pdf A for objects originating from population Ai. Let x1,x2;. .,x, be the measured values of the N objects, which are assumed to be independent. Now consider the N-tuple Gi = (gljt g2j, * * ’ 3 Sp ’ ’ ’ ...