On graphs with a constant link, II

Abstract

We study the following problem: For which finite graphs L do there exist graphs G such that the link (i.e., the neighborhood subgraph) of each vertex of G is isomorphic to L? We give a complete solution for the cases (i) L is a disjoint union of arcs, (ii) L is a tree with only one vertex of degree greater than two, (iii) L is a circle of prescribed length. Some other cases are also discussed. An interesting case is whether the situation is changed if we require G also to be finite. It transpires (see for example, Corollaries VII.3 and VII.4) that this is indeed the case.Part I of this paper will appear in [3]. It provides the basic definitions used in both part I and part II. Section III provides the basic tool, an identification procedure, that is used throughout the rest of the paper. Section IV sets up the basic building technique for the construction of more complicated graphs. It is shown how to build graphs such that the link of each vertex is an arc (of non-constant length), and how to control the proportional number of vertices with links of various lengths.