321 research outputs found
Operation of weaving partial Steiner triple systems
We introduce an operation of a kind of product which associates with a
partial Steiner triple system another partial Steiner triple system, the
starting one being a quotient of the result. We discuss relations of our
product to some other product-like constructions and prove some
preservation/non-preservation theorems. In particular, we show series of
anti-Pasch Steiner triple systems which are obtained that way
Classification of finite groups with toroidal or projective-planar permutability graphs
Let be a group. The permutability graph of subgroups of , denoted by
, is a graph having all the proper subgroups of as its vertices,
and two subgroups are adjacent in if and only if they permute. In
this paper, we classify the finite groups whose permutability graphs are
toroidal or projective-planar. In addition, we classify the finite groups whose
permutability graph does not contain one of , , , ,
or as a subgraph.Comment: 30 pages, 8 figure
Projective plane embeddings of polyhedral pinched maps
We give various conditions on pinched-torus polyhedral maps which are necessary for their graphs to be embeddable in the projective plane. Our other main result is that even if the graph of a polyhedral map in the pinched torus is embeddable in a projective plane, the map induced by the embedding cannot be polyhedral, but must have all faces bounded by cycles. Finally, we give a class of examples of graphs which have polyhedral embeddings on the pinched torus and also on orientable surfaces of arbitrary high genus
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