A simple undirected graph G is called a sum graph if there is a labeling L of the vertices of G into distinct positive integers such that any two vertices u and v of G are adjacent if and only if there is a vertex w with label L(w) = L(u) + L(v). The sum number (H) of a graph H = (V; E) is the least integer r such that graph G consisting of H and r isolated vertices is a sum graph. It is clear ...

The Wilcoxon rank sum test is frequently used in statistical practice for the comparison of measures of location when the underlying distributions are far from normal or not known in advance. An assumption of the ordinary rank sum test is that individual sampling units are independent. In many ophthalmologic clinical trials, the Early Treatment for Diabetic Retinopathy Scale (ETDRS) is a princi...

A hypergraph H is a sum hypergraph iff there are a finite S ⊆ IN and d, d ∈ IN with 1 < d ≤ d such that H is isomorphic to the hypergraph Hd,d(S) = (V, E) where V = S and E = {e ⊆ S : d ≤ |e| ≤ d ∧ ∑ v∈e v ∈ S}. For an arbitrary hypergraph H the sum number σ = σ(H) is defined to be the minimum number of isolated vertices w1, . . . , wσ 6∈ V such that H ∪ {w1, . . . , wσ} is a sum hypergraph. Fo...

Let G = (V,E) be a simple graph and for every vertex v ∈ V let L(v) be a set (list) of available colors. G is called L-colorable if there is a proper coloring φ of the vertices with φ(v) ∈ L(v) for all v ∈ V . A function f : V → N is called a choice function of G and G is said to be f -list colorable if G is L-colorable for every list assignment L with |L(v)| = f(v) for all v ∈ V . The size of ...

We introduce a variant of the well-studied sum choice number of graphs, which we call the interactive sum choice number. In this variant, we request colours to be added to the vertices’ colour-lists one at a time, and so we are able to make use of information about the colours assigned so far to determine our future choices. The interactive sum choice number cannot exceed the sum choice number ...

A simple undirected graph G is called a sum graph if there is a labeling L of the vertices of G into distinct positive integers such that any two vertices u and v of G are adjacent if and only if there is a vertex w with label L(w) = L(u) + L(v). The sum number (H) of a graph H = (V; E) is the least integer r such that graph G consisting of H and r isolated vertices is a sum graph. It is clear ...

A graph G is said to be f -choosable if there exists a proper coloring from every assignment of lists of colors to the vertices of G where the list sizes are given by f . The sum choice number of G is the minimum ∑ v∈V (G) f(v) over all f such that G is f -choosable. Here we determine the sum choice of the cartesian product P3 Pn to be 8n − 3 − bn/3c. The techniques used here have applicability...

A simple undirected graph G is called a sum graph if there exists a labelling L of the vertices of G into distinct positive integers such that any two distinct vertices u and v of G are adjacent if and only if there is a vertex w whose label L(w) = L(u) + L(v). It is obvious that every sum graph has at least one isolated vertex, namely the vertex with the largest label. The sum number (H) of a ...

We introduce a non-empirical test on pseudorandom number generators (prng), named sum-discrepancy test. We compute the distribution of the sum of consecutive m outputs of a prng to be tested, under the assumption that the initial state is uniformly randomly chosen. We measure its discrepancy from the ideal distribution, and then estimate the sample size which is necessary to reject the generato...