Chord theorems on graphs

Author:
Mohammad Javaheri

Journal:
Proc. Amer. Math. Soc. **137** (2009), 553-562

MSC (2000):
Primary 28A99; Secondary 05C99

Published electronically:
August 19, 2008

MathSciNet review:
2448575

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Abstract | References | Similar Articles | Additional Information

Abstract: The chord set of a function , denoted by , is the set of such that there exists with . It is known that if is a continuous periodic function, then it has every chord, i.e. . Equivalently, if is a real-valued Riemann-integrable function on the unit circle with , then for any , there exists an arc of length such that . In this paper, we formulate a definition of the chord set that gives way to generalizations on graphs. Given a connected finite graph , we say if for any function with there exists a connected subset of size such that . Among our results, we show that if has no vertex of degree 1, then , where is the length of the shortest closed path in . Moreover, we show that if every vertex of a connected locally finite graph has even degree, then the graph has every chord.

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Additional Information

**Mohammad Javaheri**

Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403

Address at time of publication:
Department of Mathematics, Trinity College, 300 Summit Street, Hartford, Connecticut 06106

Email:
javaheri@uoregon.edu, Mohammad.Javaheri@trincoll.edu

DOI:
https://doi.org/10.1090/S0002-9939-08-09627-5

Keywords:
Chord theorems,
Euler graphs,
chord set of locally finite graphs

Received by editor(s):
January 22, 2008

Published electronically:
August 19, 2008

Communicated by:
Jim Haglund

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.