Chord theorems on graphs

The Horizontal Chord Theorem states that if a continuous curve connects points A and B in the plane, then for any integer k there are points C and D on the curve such that −−→ AB = k −−→ CD. In this note, we discuss a few combinatorial-analysis problems related to this theorem and introduce a different formulation that gives way to generalizations on graphs. 1 THE NECKLACE OF PEARLS PROBLEM. Two pirates have a single-strand necklace containing 2N black pearls and 2N white pearls arranged in any order. They would like to cut the necklace into as few pieces as possible so that after dividing the pieces of the necklace between them, each gets exactly N white pearls and N black ones. Using the following theorem, one can show that two cuts are sufficient. V. Totik has discussed several problems, including the above mentioned and the combinatorial-analysis theorem below, in the Monthly paper ‘A Tale of Two Integrals’ [4].