Some Results on Chromaticity of Quasilinear Hypergraphs

Abstract

Let r ≥ 1 be an integer. An h-hypergraph H is said to be r-quasilinear (linear for r = 1) if any two edges of H intersect in 0 or r vertices. This paper surveys chromaticity of r-quasilinear h-hypergraphs with emphasis to sunflower hypergraphs SH(n, p, h), r-quasilinear paths P _m ^h, r of length m ≥ 1, and cycles C _m ^h, r of length m ≥ 3. They are chromatically unique in the set of h-uniform hypergraphs provided 1 ≤ p ≤ h − 2, or $p = h - 1$ and they have at most two edges, and in the set of h-uniform r-quasilinear hypergraphs if r ≥ 2 and h ≥ 3r − 1, respectively.