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- Title
Edge balanced star‐hypergraph designs and vertex colorings of path designs.
- Authors
Bonacini, Paola; Marino, Lucia
- Abstract
Let Kv(3)=(X,ℰ) ${K}_{v}^{(3)}=(X,{\rm{ {\mathcal E} }})$ be the complete hypergraph, uniform of rank 3, defined on a vertex set X={x1,...,xv} $X=\{{x}_{1},\ldots ,{x}_{v}\}$, 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 Kv(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+43 ${S}^{(3)}\left(2,\frac{v+4}{3}\right)$, with v≡2mod3 $v\equiv 2\,\mathrm{mod}\,3$ sufficiently big, for any p∈N $p\in {\mathbb{N}}$, v2≤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 2v $2v$ 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.
- Subjects
COLOR in design; RAMSEY numbers; GRAPH coloring; EDGES (Geometry)
- Publication
Journal of Combinatorial Designs, 2022, Vol 30, Issue 7, p497
- ISSN
1063-8539
- Publication type
Article
- DOI
10.1002/jcd.21837