On $H$-Chromatic Uniqueness of Two Linear Uniform Cycles Having Some Edges in Common

Abstract

In this paper, it is proved that the $h$-chromatic uniqueness of the linear $h$-hypergraph consisting of two cycles of lengths $p$ and $q$ having $r$ edges in common when $p=q$, $2\le r\le p-2$, and $h\ge 3$. We also obtain the chromatic polynomial of a connected unicyclic linear $h$-hypergraph and show that every $h$-uniform cycle of length three is not chromatically unique for $h\ge 3$.