2 research outputs found
Distinguishing partitions of complete multipartite graphs
A \textit{distinguishing partition} of a group with automorphism group
is a partition of that is fixed by no nontrivial element of
. In the event that 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
is a complete multipartite graph with parts of size and
parts of size for small , and large , . A key tool
in making the estimate is counting the number of trees of particular classes
Connection Matrices and the Definability of Graph Parameters
In this paper we extend and prove in detail the Finite Rank Theorem for
connection matrices of graph parameters definable in Monadic Second Order Logic
with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and
J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying
known and new non-definability results of graph properties and finding new
non-definability results for graph parameters. We also prove a Feferman-Vaught
Theorem for the logic CFOL, First Order Logic with the modular counting
quantifiers