Chord theorems on graphs

Author:
Mohammad Javaheri

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

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

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

Published electronically:
August 19, 2008

MathSciNet review:
2448575

Full-text PDF Free Access

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.

**1.***Contests in Higher Mathematics*, 1949-1961, Akadémiai Kiadó, Budapest, 1968. MR**0239895 (39:1252)****2.**J.P. Huneke,*Mountain Climbing*, Trans. Amer. Math. Soc.**139**(1969) 383-391. MR**0239013 (39:372)****3.**J.C. Oxtoby,*Horizontal Chord Theorem*, Amer. Math. Monthly**79**(1972) 468-475. MR**0299735 (45:8783)****4.**K.A. Ross,*Elementary Analysis: The Theory of Calculus*, Springer-Verlag, 1980. MR**560320 (81a:26001)****5.**V. Totik,*A Tale of Two Integrals*, Amer. Math. Monthly**106**(1999) 227-240. MR**1682343 (2000d:26002)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
28A99,
05C99

Retrieve articles in all journals with MSC (2000): 28A99, 05C99

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.