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
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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.
 1.
Contests in higher mathematics (Hungary, 1949–1961), In
memoriam Miklós Schweitzer. Editorial Board: G. Szász, L.
Gehér, I. Kovács and L. Pintér. Manuscript revised by
P. Erdös, A. Rényi, B. Sz.Nagy and P. Turán.
Linguistically revised by B. Balkay, Akadémiai Kiadó,
Budapest, 1968. MR 0239895
(39 #1252)
 2.
John
Philip Huneke, Mountain climbing, Trans. Amer. Math. Soc. 139 (1969), 383–391. MR 0239013
(39 #372), http://dx.doi.org/10.1090/S00029947196902390139
 3.
J.
C. Oxtoby, Horizontal chord theorems, Amer. Math. Monthly
79 (1972), 468–475. MR 0299735
(45 #8783)
 4.
Kenneth
A. Ross, Elementary analysis: the theory of calculus,
SpringerVerlag, New York, 1980. Undergraduate Texts in Mathematics. MR 560320
(81a:26001)
 5.
Vilmos
Totik, A tale of two integrals, Amer. Math. Monthly
106 (1999), no. 3, 227–240. MR 1682343
(2000d:26002), http://dx.doi.org/10.2307/2589678
 1.
 Contests in Higher Mathematics, 19491961, Akadémiai Kiadó, Budapest, 1968. MR 0239895 (39:1252)
 2.
 J.P. Huneke, Mountain Climbing, Trans. Amer. Math. Soc. 139 (1969) 383391. MR 0239013 (39:372)
 3.
 J.C. Oxtoby, Horizontal Chord Theorem, Amer. Math. Monthly 79 (1972) 468475. MR 0299735 (45:8783)
 4.
 K.A. Ross, Elementary Analysis: The Theory of Calculus, SpringerVerlag, 1980. MR 560320 (81a:26001)
 5.
 V. Totik, A Tale of Two Integrals, Amer. Math. Monthly 106 (1999) 227240. MR 1682343 (2000d:26002)
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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.
