Edge balanced star‐hypergraph designs and vertex colorings of path designs

Let K v ( 3 ) = ( X , ℰ ) ${K}_{v}^{(3)}=(X,{\rm{ {\mathcal E} }})$ be the complete hypergraph, uniform of rank 3, defined on a vertex set X = { x 1 , … , x v } $X=\{{x}_{1},\ldots ,{x}_{v}\}$ , so that ℰ ${\rm{ {\mathcal E} }}$ is the set of all triples of X $X$ . Let H ( 3 ) = ( V , D ) ${H}^{(3)}=(V,{\mathscr{D}})$ be a subhypergraph of K v ( 3 ) ${K}_{v}^{(3)}$ , which means that V ⊆ X $V\subseteq X$ and D ⊆ ℰ ${\mathscr{D}}\subseteq {\rm{ {\mathcal E} }}$ . We call 3‐edges the triples of V $V$ contained in the family D ${\mathscr{D}}$ and edges the pairs of V $V$ contained in the 3‐edges of D ${\mathscr{D}}$ , that we denote by [ x , y ] $[x,y]$ . A H ( 3 ) ${H}^{(3)}$ ‐design Σ ${\rm{\Sigma }}$ is called edge balanced if for any x , y ∈ X $x,y\in X$ , x ≠ y $x\ne y$ , the number of blocks of Σ ${\rm{\Sigma }}$ containing the edge [ x , y ] $[x,y]$ is constant. In this paper, we consider the star hypergraph S ( 3 ) ( 2 , m + 2 ) ${S}^{(3)}(2,m+2)$ , which is a hypergraph with m $m$ 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 ) ${S}^{(3)}(2,m+2)$ ‐designs for any m ≥ 2 $m\ge 2$ , that is, the set of the orders v $v$ for which such a design exists. Then we consider the case m = 2 $m=2$ and we denote the hypergraph S ( 3 ) ( 2 , 4 ) ${S}^{(3)}(2,4)$ by P ( 3 ) ( 2 , 4 ) ${P}^{(3)}(2,4)$ . Starting from any edge‐balanced S ( 3 ) 2 , v + 4 3 ${S}^{(3)}\left(2,\frac{v+4}{3}\right)$ , with v ≡ 2 mod 3 $v\equiv 2\,\mathrm{mod}\,3$ sufficiently big, for any p ∈ N $p\in {\mathbb{N}}$ , v 2 ≤ p ≤ v $\unicode{x02308}\frac{v}{2}\unicode{x02309}\le p\le v$ , we construct a P ( 3 ) ( 2 , 4 ) ${P}^{(3)}(2,4)$ ‐design of order 2 v $2v$ with feasible set { 2 , 3 } ∪ [ p , v ] $\{2,3\}\cup [p,v]$ , in the context of proper vertex colorings such that no block is either monochromatic or polychromatic.