Some Results on Chromaticity of Quasi-Linear Paths and Cycles

Abstract

Let $r\geq 1$ be an integer. An $h$-hypergraph $H$ is said to be $r$-quasi-linear (linear for $r=1$) if any two edges of $H$ intersect in 0 or $r$ vertices. In this paper it is shown that $r$-quasi-linear paths $P_{m}^{h,r}$ of length $m\geq 1$ and cycles $C_{m}^{h,r}$ of length $m\geq 3$ are chromatically unique in the set of $h$-uniform $r$-quasi-linear hypergraphs provided $r\geq 2$ and $h\geq 3r-1$.

(k − 2)-linear connected components in hypergraphs of rank k

“…We thus introduce the general concept of q-linear path, where any two consecutive edges intersect on between 1 and q vertices (and non-consecutive edges do not intersect). Extremal results also exist on paths with similar restrictions on the size of the intersections, for example paths where any two consecutive edges must intersect on exactly q vertices [Tomescu (2012)] [Dudek et al (2017)] with emphasis on the linear case q = 1 [Füredi et al (2014)] [Gu et al (2020)]. Throughout this article, let H be a hypergraph of rank k: as for any hypergraph, we denote its vertex set by V (H) and its edge set by E(H).…”