Sachs' linkless embedding conjecture
The core part of this paper is a proof of Sachs' conjecture, which states that a graph can be linklessly embedded in the 3-space iff it has no Petersen family minor, i.e., iff it contains as minor none of the seven graphs obtainable from \(K_6\) by \(Y-\Delta\) and \(\Delta- Y\) exchanges. Other results: (1) A proof of a conjecture of Böhme stating that a graph admits a linkless embedding iff it admits a panelled embedding (one in which every circuit of the graph is the boundary of a disc disjoint from the remainder of the graph). (2) An embedding is panelled iff the complement in 3-space of every subgraph has free fundamental group. This extends a theorem of Scharelmann and Thompson for planar graphs. (3) If two panelled embeddings are not related by an orientation-preserving homeomorphism of the 3-space, then there is a subgraph which is a subdivision of \(K_5\) or \(K_{3,3}\) such that the two embeddings of this subgraph are still ``different''.
scientific article; zbMATH DE number 798747