Chord theorems on graphs
Author:
Mohammad Javaheri
Journal:
Proc. Amer. Math. Soc. 137 (2009), 553562
MSC (2000):
Primary 28A99; Secondary 05C99
Published electronically:
August 19, 2008
MathSciNet review:
2448575
Fulltext PDF Free Access
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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 realvalued Riemannintegrable 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:
http://dx.doi.org/10.1090/S0002993908096275
PII:
S 00029939(08)096275
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.
