In 1995, Stiebitz [13] asked the following question: For any positive integers s, t, is there a finite integer f(s, t) such that every digraph D with minimum out-degree at least f(s, t) admits a bipartition (A,B) such that A induces a subdigraph with minimum out-degree at least s and B induces a subdigraph with minimum out-degree at least t? We give an affirmative answer for tournaments, bipart...

A new Joining and model Checking scheme (JaCk-SAT) for parallel resolution of the SATisfiability problem is presented. The main goal of this methodology is to cut the variable-set in two subsets of about equal size. In contrast with recent propositions [12,16] for sequential resolution only, we do not use sophisticated hypergraph decomposition techniques such as Tree Decomposition that are very...

Let A = (ai,j) , i = 1, 2, . . . , j = 0, 1, 2, . . . , be an infinite matrix with elements ai,j = 0 or 1; p (n, k;A) the number of partitions of n into k parts whose number yi of parts which are equal to i belongs to the set Yi = {j : ai,j = 1} , i = 1, 2, . . . . The universal theorem on partitions states that 1 X n=0 n X k=0 p (n, k;A)ut = 1 Y i=1 0 @ 1 X j=0 ai,ju t 1 A . In this paper, we ...

Let Γ be a finite locally (G, s)-arc transitive graph with s ≥ 2 such that G is intransitive on vertices. Then Γ is bipartite and the two parts of the bipartition are G-orbits. In previous work the authors showed that if G has a nontrivial normal subgroup intransitive on both of the vertex orbits of G, then Γ is a cover of a smaller locally s-arc transitive graph. Thus the “basic” graphs to stu...

Given a quasi-ordering of labels, a labelled ordered tree s is embedded with gaps in another tree t if there is an injection from the nodes of s into those of t that maps each edge in s to a unique disjoint path in t with greater-or-equivalent labels, and which preserves the order of children. We show that finite trees are well-quasiordered with respect to gap embedding when labels are taken fr...