Interpolation theorem for a continuous function on orientations of a simple graph

Fuji Zhang, Zhibo Chen

Research output: Contribution to journalArticle

Abstract

Let G be a simple graph. A function f from the set of orientations of G to the set of non-negative integers is called a continuous function on orientations of G if, for any two orientations O1 and O2 of G, |f(O1) - f(O2)| ≤ 1 whenever O1 and O2 differ in the orientation of exactly one edge of G. We show that any continuous function on orientations of a simple graph G has the interpolation property as follows: If there are two orientations O1 and O2 of G with f(O1) = p and f(O2) = q, where p < q, then for any integer k such that p < k < q, there are at least m orientations O of G satisfying f(O) = k, where m equals the number of edges of G. It follows that some useful invariants of digraphs including the connectivity, the arcconnectivity and the absorption number, etc., have the above interpolation property on the set of all orientations of G.

Original languageEnglish (US)
Pages (from-to)433-438
Number of pages6
JournalCzechoslovak Mathematical Journal
Volume48
Issue number3
DOIs
StatePublished - Jan 1 1998

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Simple Graph
Continuous Function
Interpolate
Theorem
Integer
Digraph
Connectivity
Absorption
Non-negative
Invariant

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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abstract = "Let G be a simple graph. A function f from the set of orientations of G to the set of non-negative integers is called a continuous function on orientations of G if, for any two orientations O1 and O2 of G, |f(O1) - f(O2)| ≤ 1 whenever O1 and O2 differ in the orientation of exactly one edge of G. We show that any continuous function on orientations of a simple graph G has the interpolation property as follows: If there are two orientations O1 and O2 of G with f(O1) = p and f(O2) = q, where p < q, then for any integer k such that p < k < q, there are at least m orientations O of G satisfying f(O) = k, where m equals the number of edges of G. It follows that some useful invariants of digraphs including the connectivity, the arcconnectivity and the absorption number, etc., have the above interpolation property on the set of all orientations of G.",
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Interpolation theorem for a continuous function on orientations of a simple graph. / Zhang, Fuji; Chen, Zhibo.

In: Czechoslovak Mathematical Journal, Vol. 48, No. 3, 01.01.1998, p. 433-438.

Research output: Contribution to journalArticle

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