A graph G = (V,E) with p vertices and q edges is said to be a total mean cordial graph if there exists a function f : V (G) → {0, 1, 2} such that f(xy) = [(f(x)+f(y))/2] where x, y ∈ V (G), xy ∈ E(G), and the total number of 0, 1 and 2 are balanced. That is |evf (i) − evf (j)| ≤ 1, i, j ∈ {0, 1, 2} where evf (x) denotes the total number of vertices and edges labeled with x (x = 0, 1, 2). In thi...

A Skolem sequence is a sequence a1, a2, . . . , a2n (where ai ∈ A = {1, . . . , n}), each ai occurs exactly twice in the sequence and the two occurrences are exactly ai positions apart. A set A that can be used to construct Skolem sequences is called a Skolem set. The existence question of deciding which sets of the form A = {1, . . . , n} are Skolem sets was solved by Thoralf Skolem [6] in 195...

It was shown by Abrham that the number of pure Skolem sequences of order n, n ≡ 0 or 1 (mod 4), and the number of extended Skolem sequences of order n, are both bounded below by 2bn/3c. These results are extended to give similar lower bounds for the numbers of hooked Skolem sequences, split Skolem sequences and split-hooked Skolem sequences.

a graph g = (v,e) with p vertices and q edges is said to be a total mean cordial graph if there exists a function f : v (g) → {0, 1, 2} such that f(xy) = [(f(x)+f(y))/2] where x, y ∈ v (g), xy ∈ e(g), and the total number of 0, 1 and 2 are balanced. that is |evf (i) − evf (j)| ≤ 1, i, j ∈ {0, 1, 2} where evf (x) denotes the total number of vertices and edges labeled with x (x = 0, 1, 2). in thi...

In this paper, we consider Skolem (vertex) labellings and present (hooked) Skolem labellings for generalised Dutch windmills whenever such labellings exist. Specifically, we show that generalised Dutch windmills with more than two cycles cannot be Skolem labelled and that those composed of two cycles of lengths m and n, n ≥ m, cannot be Skolem labelled if and only if n−m ≡ 3, 5 (mod 8) and m is...

We consider the problem of computing Skolem functions for satisfied dependency quantified Boolean formulas (DQBFs). We show how Skolem functions can be obtained from an elimination-based DQBF solver and how to take preprocessing steps into account. The size of the Skolem functions is optimized by don’t-care minimization using Craig interpolants and rewriting techniques. Experiments with our DQB...