11-Doilies with Vertex Sets of Sizes 275, 286, … , 462

Abstract

In this paper, we close the gap between the results of [8] and [9] by showing that there are 11 -doilies for vertex sets V of size |V| = 275, 286,…,462. In [8] it is shown that there are 11 -doilies with |V| = 462, 473,…, 1001, and in [9] it is shown that there are such diagrams with |V| = 231, 242,…, 352. This is the third paper in a series intended to show that 11 -doilies exist for any possible size of vertex set, answering a question of Grünbaum. We continue to extend the method that is developed in [7], [8], and [9]. The crucial step in the method of those papers is based on saturated chain decompositions of planar spanning subgraphs of the p -hypercube, and on edge-disjoint path decompositions of planar spanning subgraphs of the p -hypercube. We continue to study these types of decompositions. In constructing the doilies we use a basic structure called the Venn graph of a doodle of the doily. Using the new Venn graph that we create here we show the existence of the above mentioned diagrams. In fact, without checking all the details we show that there are at least 217 (non-isomorphic) 11 -doilies with these sizes of vertex sets.