Given an integer k ≥ 2, we consider vertex colorings of graphs in which no k-star subgraph Sk = K1,k is polychromatic. Equivalently, in a star-[k]-coloring the closed neighborhood N[v] of each vertex v can have at most k different colors on its vertices. The maximum number of colors that can be used in a star-[k]-coloring of graph G is denoted by χ̄k⋆(G) and is termed the star-[k] upper chromati...

This paper considers quasi-reduced multi-valued decision diagrams with k bits (QRMDD(k)s) to represent twovalued logic functions. It shows relations between the numbers of nodes in QRMDD(k)s and values of k for benchmark functions; an upper bound on the number of nodes in the QRMDD(k); difference between the upper bound and the number of nodes in the QRMDD(k)s for random functions; and the amou...

A k-decomposition (G1, . . . , Gk) of a graph G is a partition of its edge set to form k spanning subgraphs G1, . . . , Gk. The classical theorem of Nordhaus and Gaddum bounds χ(G1) + χ(G2) and χ(G1)χ(G2) over all 2-decompositions of Kn. For a graph parameter p, let p(k;G) denote the maximum of ∑k i=1 p(Gi) over all k-decompositions of the graph G. The clique number ω, chromatic number χ, list ...

For any number field k, upper bounds are established for the number of k-rational points of bounded height on non-singular del Pezzo surfaces defined over k, which are equipped with suitable conic bundle structures over k.

A nonempty set S ⊆ V is a defensive k-alliance in Γ = (V,E), k ∈ [−Δ,Δ] ∩ Z, if for every v ∈ S, δS(v) ≥ δS̄(v) + k. An defensive kalliance S is called critical if no proper subset of S is an defensive k-alliances in Γ = (V,E). The upper defensive k-alliance number of Γ, denoted by Ak(Γ), is defined as the maximum cardinality of a critical defensive k-alliance in Γ. In this paper we study the ma...

An Sn,k refers to a generalized version of an n-star graph (Sn), where Sn,n−1 and Sn are isomorphic and Sn,1 is obviously a complete graph of n vertices (Kn). This study constructs a set of many-to-one vertex-disjoint paths in an (n, k)-star graph (Sn,k), and computes the upper bound of Rabin number of an Sn,k, where 2≤k≤n−1. The upper bound of the Rabin number of an Sn,k is calculated by const...

We study fractional scheduling problems in sensor networks, in particular, sleep scheduling (generalisation of fractional domatic partition) and activity scheduling (generalisation of fractional graph colouring). The problems are hard to solve in general even in a centralised setting; however, we show that there are practically relevant families of graphs where these problems admit a local dist...

Sharp exponential upper bound, k!n−1, on the number of hamiltonian k-sets (i.e., decompositions into k hamiltonian cycles) among multigraphs G is found if the number, n, of vertices is fixed, n ≥ 3. Moreover, the upper bound is attained iff G = Cn where Cn is the k-fold n-cycle Cn. Furthermore, if G 6= Cn then the number of hamiltonian k-sets in G is less than or equal to k!n−1/k, the bound, if...

Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number γ×k(G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V , |NG[v]∩S| ≥ k. Also the total k-domination number γ×k,t(G) of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V , |NG(v)∩S| ≥ k. The k-transversal numb...