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

Chord theorems on graphs

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
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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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