TMC tetrahedral types MOD \(2k+1\) and their structure graphs
This paper refers to several previous papers where the author has considered edge-weighted tetrahedra with a zero-sum condition on their triangles as the vertices of a certain graph whose adjacency is given by the sharing of pairs of edge-weighted triangles. These edge-weighted tetrahedra are also viewed as totally multicolored (TMC) \(K_4\) subgraphs of a certain Cayley graph, which is an infinite complete graph whose edges are colored (weighted) with positive integers. (Here TMC means that the edges of the subgraph have all distinct colors.) In the present paper the situation is considered modulo an odd integers \(n = 2k + 1\), so that the entire graph of tetrahedra is finite (and its vertices corresponds to TMC subgraphs of a finite complete graph \(K_n)\). Here the edge-weights are taken as 0 through \(k\), treated as the absolute values of integers modulo \(n\) (the author uses ``\(\text{MOD} n\)'' to distinguish this from the usual ``\(\bmod n\)''). Several aspects are discussed of the resulting structure graph of tetrahedra, and in particular it is proved to be an edge-disjoint union of triangles (with at most three of the triangles meeting at any one vertex), except for possible isolated vertices. Examples for several values of \(n\) are described in detail, and for general \(n\) connections with the infinite covering graph and some of its subgraphs are also discussed. These include some coverings which can be viewed as regular tesselations of a plane by hexagons and/or triangles.
scientific article; zbMATH DE number 887176
scientific article