TMC tetrahedral types MOD \(2k+1\) and their structure graphs
type: Item
instance of: scholarly article
title: TMC tetrahedral types MOD \(2k+1\) and their structure graphs
zbMATH Open document ID: 0856.05038
DOI: 10.1007/BF01858451
published in: Graphs and Combinatorics
publication date: 1996-08-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 05B45
zbMATH DE Number: 887176
zbMATH Keywords: edge-weighted triangles
zbMATH Keywords: edge-weighted tetrahedra
zbMATH Keywords: Cayley graph
zbMATH Keywords: covering graph
zbMATH Keywords: tesselations
author: Italo J. Dejter
reviewed by: Albert J. Goodman
MaRDI profile type: Publication
cites work: The classification of minimal graphs with given abelian automorphism group
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cites work: HOW TO DESCRIBE A GRAPH
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description: scientific article; zbMATH DE number 887176