On Perfect Balanced Rainbow-Free Colorings and Complete Colorings of Projective Spaces
Bulletin of the Malaysian Mathematical Sciences Society(2024)
Abstract
This paper is motivated by the problem of determining the related chromatic numbers of some hypergraphs. A hypergraph Π _q(n,k) is defined from a projective space PG (n-1,q) , where the vertices are points and the hyperedges are (k-1) -dimensional subspaces. For the perfect balanced rainbow-free colorings, we show that χ_p(Π _q(n,k))=q^n-1/l(q-1) , where k≥⌈n+1/2⌉ and l is the smallest nontrivial factor of q^n-1/q-1 . For the complete colorings, we prove that there is no complete coloring for Π _q(n,k) with 2≤ k
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Key words
Hypergraph,Projective space,Coloring,Perfect balanced rainbow-free coloring,Complete coloring,05C15,05B25,51E20
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