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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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: The chord set of a function $ f: \mathbb{R} \rightarrow \mathbb{R}$, denoted by $ H(f)$, is the set of $ r\in \mathbb{R}$ such that there exists $ x\in \mathbb{R}$ with $ f(x+r)=f(x)$. It is known that if $ f$ is a continuous periodic function, then it has every chord, i.e. $ H(f)=\mathbb{R}$. Equivalently, if $ f$ is a real-valued Riemann-integrable function on the unit circle $ C$ with $ \int_C f =0$, then for any $ r\in [0,1]$, there exists an arc $ L$ of length $ r$ such that $ \int_L f=0$. In this paper, we formulate a definition of the chord set that gives way to generalizations on graphs. Given a connected finite graph $ G$, we say $ r\in H(G)$ if for any function $ f \in L^1(G)$ with $ \int_G f=0$ there exists a connected subset $ A$ of size $ r$ such that $ \int_A f=0$. Among our results, we show that if $ G$ has no vertex of degree 1, then $ [0,l(G)] \subseteq H(G)$, where $ l(G)$ is the length of the shortest closed path in $ G$. 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

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.