Representation of finite graphs as difference graphs of S-units. II

K. Györy, L. Hajdu, R. Tijdeman

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Abstract

In Part I of the present paper the following problem was investigated. Let G be a finite simple graph, and S be a finite set of primes. We say that G is representable with S if it is possible to attach rational numbers to the vertices of G such that the vertices v1, v2 are connected by an edge if and only if the difference of the attached values is an S-unit. In Part I we gave several results concerning the representability of graphs in the above sense. In the present paper we extend the results from Part I to the algebraic number field case and make some of them effective. Besides we prove some new theorems: we prove that G is infinitely representable with S if and only if it has a degenerate representation with S, and we also deal with the representability with S of the union of two graphs of which at least one is finitely representable with S.

Original languageEnglish
Pages (from-to)423-447
Number of pages25
JournalActa Mathematica Hungarica
Volume149
Issue number2
DOIs
Publication statusPublished - Aug 1 2016

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Keywords

  • arithmetic graph
  • cubical graph
  • representability
  • S-unit equation

ASJC Scopus subject areas

  • Mathematics(all)

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