Automorphism groups and the full state spaces of the Petersen graph generalizations of G32

AbstractThe geometric duals of the generalized Petersen graphs G(n, k) are the Greechie representations of the Generalizations of G32. The duals are denotes by G∗(n, k) and the generalizations by L(G∗(n, k)). For these generalizations which are orthomodular posets and lattices, the automorphism groups are completely determined. State properties are also investigated with the following results obtaining. 1.(1) L(G∗(n, 1)) admits a full set of dispersion free states if n is even.2.(2) L(G∗(n, 1)) does not admit a full set of states if n is odd.3.(3) L(G∗(n, 2)) admits a full set of dispersion free states for all values of n other than 5 or 8.4.(4) L(G∗(8, 2)) admits a full set of states but does not admit a full set of dispersion free states.5.(5) L(G∗(5, 2)) does not admit a full set of states.6.(6) L(G∗(n, 3)) admits a full set of dispersion free states for all n

Combinatorial Embeddings and Representations

Topological embeddings of complete graphs and complete multipartite graphs give rise to combinatorial designs when the faces of the embeddings are triangles. In this case, the blocks of the design correspond to the triangular faces of the embedding. These designs include Steiner, twofold and Mendelsohn triple systems, as well as Latin squares. We look at construction methods, structural properties and other problems concerning these cases. In addition, we look at graph representations by Steiner triple systems and by combinatorial embeddings. This is closely related to finding independent sets in triple systems. We examine which graphs can be represented in Steiner triple systems and combinatorial embeddings of small orders and give several bounds including a bound on the order of Steiner triple systems that are guaranteed to represent all graphs of a given maximum degree. Finally, we provide an enumeration of graphs of up to six edges representable by Steiner triple systems