A distinguishing partition of a set X with automorphism group aut(X) is a partition of X that is fixed by no nontrivial element of aut(X). In the event that X is a complete multipartite graph with its automorphism group, the existence of a distinguishing partition is equivalent to the existence of an asymmetric hypergraph with prescribed edge sizes. An asymptotic result is proven on the existence of a distinguishing partition when X is a complete multipartite graph with m1 parts of size n1 and m2 parts of size n2 for small n1, m2 and large m1, n2. A key tool in making the estimate is counting the number of trees of particular classes.
@article{428,
title = {Distinguishing partitions of complete multipartite graphs},
journal = {ARS MATHEMATICA CONTEMPORANEA},
volume = {9},
year = {2014},
doi = {10.26493/1855-3974.428.296},
language = {EN},
url = {http://dml.mathdoc.fr/item/428}
}
Goff, Michael. Distinguishing partitions of complete multipartite graphs. ARS MATHEMATICA CONTEMPORANEA, Tome 9 (2014) . doi : 10.26493/1855-3974.428.296. http://gdmltest.u-ga.fr/item/428/