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

K. Györy, L. Hajdu, R. Tijdeman

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 language English 423-447 25 Acta Mathematica Hungarica 149 2 https://doi.org/10.1007/s10474-016-0633-y Published - Aug 1 2016

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Representability
Finite Graph
If and only if
Algebraic number Field
Unit
Graph in graph theory
Simple Graph
Finite Set
Union
Theorem

Keywords

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

ASJC Scopus subject areas

• Mathematics(all)

Cite this

Representation of finite graphs as difference graphs of S-units. II. / Györy, K.; Hajdu, L.; Tijdeman, R.

In: Acta Mathematica Hungarica, Vol. 149, No. 2, 01.08.2016, p. 423-447.

Research output: Contribution to journalArticle

Györy, K. ; Hajdu, L. ; Tijdeman, R. / Representation of finite graphs as difference graphs of S-units. II. In: Acta Mathematica Hungarica. 2016 ; Vol. 149, No. 2. pp. 423-447.
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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.",
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N2 - 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.

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