On the distinguishing partitions and asymmetric uniform hypergraphs

This study is an exposition of the first three sections of the paper entitled Distinguishing Partitions and Asymmetric Uniform Hypergraphs by Ellingham and Schroeder, which appeared in ARS Mathematica Comtemporanea [7]. We give a thorough discussion of the preliminary concepts, proofs of propositions, theorems, and lemmas found in the paper. We also give a discussion on some important properties of hypergraphs. Further, we determine when a distinguishing partition for some special graphs and asymmetric hypergraphs exists. Also, we provided a lemma which states that there are no asymmetric 2-uniform hypergraphs with edges 1 m 5. Lastly, we present our observation that having exactly one nontrivial automorphism in its automorphism group, a graph with 2-distinguishing coloring has a distinguishing partition if and only if there exists a vertex v 2 V (G) such that 2(v) = v