# 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 language English (US) 433-438 6 Czechoslovak Mathematical Journal 48 3 https://doi.org/10.1023/A:1022471626622 Published - Jan 1 1998

### Fingerprint

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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T1 - Interpolation theorem for a continuous function on orientations of a simple graph

AU - Zhang, Fuji

AU - Chen, Zhibo

PY - 1998/1/1

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N2 - 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.

AB - 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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