Two-Chord Framings of Spanning Trees

Abstract

We find sufficient conditions under which a finite connected graph has a spanning tree with the following property. There is a numbering of edges and an injective mapping of the set of all edges of the tree to the set of all pairs of different chords (i.e., edges of the graph not contained in the tree) such that for any pair of chords in the image of the mapping, the cycles containing one chord from the pair and containing no other chords intersect along an edge in the preimage, and, maybe, along other edges of the tree with smaller numbers. The problem of study of graphs that possess this property appeared in the process of study the (isotopic) classification problem of embeddings of graphs in the 3-space. Bibliography: 3 titles.