CHORD THEOREMS ON GRAPHS

The chord set of a function f : R -> R, denoted by H(f), is the set of r epsilon R such that there exists x epsilon 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) = R. Equivalently, if f is a real-valued Riemann-integrable function on the unit circle C with integral(C) f = 0, then for any r epsilon [ 0, 1], there exists an arc L of length r such that integral(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 epsilon H(G) if for any function f epsilon L(1)(G) with integral(G) f = 0 there exists a connected subset A of size r such that integral(A) f = 0. Among our results, we show that if G has no vertex of degree 1, then [ 0, l(G)] subset of 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.