The domatic number of a graph G, denoted dom(G), is the maximum possible cardinality of a family of disjoint sets of vertices of G, each set being a dominating set of G. It is well known that every graph without isolated vertices has dom(G) ≥ 2. For every k, it is known that there are graphs with minimum degree at least k and with dom(G) = 2. In this paper we prove that this is not the case if ...

We investigate the theoretical feasibility of near-optimal, distributed sleep scheduling in energyconstrained sensor networks with pairwise sensor redundancy. In this setting, an optimal sleep schedule is equivalent to an optimal fractional domatic partition of the associated redundancy graph. We present a set of realistic assumptions on the structure of the communication and redundancy relatio...

For k ≥ 1, we consider interleaved k-tuple colorings of the nodes of a graph, that is, assignments of k distinct natural numbers to each node in such a way that nodes that are connected by an edge receive numbers that are strictly alternating between them with respect to the relation <. If it takes at least χint(G) distinct numbers to provide graph G with such a coloring, then the interleaved m...

Top-k query is an efficient way to show the most important objects to user from massive amounts of data. After huge effort working on database performance, recently, an explain capability has attract more attention in recent years. In top-k query, since people may not specify his/her accurate preference, he may feel frustrated with the top-k results and propose a question such that “why my expe...

In order to update k-anonymous and confidential database, the suppression based and generalization based updating protocol technique has been proposed. These protocols check whether the database inserted with the new tuple is still k anonymous without knowing the content of the table and database respectively. But these methods will not work if initial database is empty. Also, if the incoming t...

For fixed non-negative integers $k$, $t$, and $n$, with $t &lt; k$, a $k_t$-Dyck path of length $(k+1)n$ is lattice that starts at $(0, 0)$, ends $((k+1)n, stays weakly above the line $y = -t$, consists steps from step-set $\{(1, 1), (1, -k)\}$. We enumerate family paths by considering number down-steps height $i$ modulo $k$. Given tuple $(a_1, a_2, \ldots, a_k)$ we find an exact enumeration fo...

We obtain a complete characterization for doubly commuting mixed invariant subspaces of the Hardy space over unit polydisc. say closed subspace Q H2(Dn) is if Mzj(Q)⊆Q 1≤j≤k and Mzj⁎(Q)⊆Q, k+1≤j≤n some integer k∈{1,2,…,n−1}. prove that only ifQ=ΘH2(Dk)⊗Qθ1⊗⋯⊗Qθn−k, where Θ∈H∞(Dk) inner function Qθj either Jordan block H2(D)⊖θjH2(D) θj or H2(D). Furthermore, an explicit representation commutant ...