A note on asymmetric hypergraphs

Abstract

A $k$-graph $\mathcal{G}$ is asymmetric if there does not exist an automorphism on $\mathcal{G}$ other than the identity, and $\mathcal{G}$ is called minimal asymmetric if it is asymmetric but every non-trivial induced sub-hypergraph of $\mathcal{G}$ is non-asymmetric. Extending a result of Jiang and Nešetřil, we show that for every $k$-graph, $k\ge 3$, there exist infinitely many minimal asymmetric $k$-graphs which have maximum degree $2$ and are linear. Further, we show that there are infinitely many $2$-regular asymmetric $k$-graphs for $k\ge 3$.

9 pages, 5 figures