Edge Balanced Star‐hypergraph Designs and Vertex Colorings of Path Designs

Let K-v((3)) = (X, E) be the complete hypergraph, uniform of rank 3, defined on a vertex set X = {x(1), ..., x(v)}, so that E is the set of all triples of X. Let H-(3) = (V, D) be a subhypergraph of K-v((3)), which means that V subset of X and D subset of E. We call 3-edges the triples of V contained in the family. and edges the pairs of V contained in the 3-edges of V, that we denote by [x, y]. A H-(3)-design Sigma is called edge balanced if for any x, y is an element of X, x not equal y, the number of blocks of Sigma containing the edge [x, y] is constant. In this paper, we consider the star hypergraph S-(3)(2, m + 2), which is a hypergraph with m 3-edges such that all of them have an edge in common. We completely determine the spectrum of edge balanced S-(3)(2, m + 2)-designs for any m >= 2, that is, the set of the orders v for which such a design exists. Then we consider the case m = 2 and we denote the hypergraph S-(3)(2, 4) by P-(3)(2, 4). Starting from any edge-balanced S-(3)(2, v + 4/3), with v equivalent to 2 mod 3 sufficiently big, for any p is an element of N, [v/2] <= p <= v, we construct a P-(3)(2, 4)-design of order 2(v) with feasible set {2, 3} boolean OR [p, v], in the context of proper vertex colorings such that no block is either monochromatic or polychromatic.