A graph theoretic proof of sharkovsky’s theorem on the periodic points of continuous functions

Abstract

Let f be a continuous real valued function defined on the real line. If f has a periodic point of period k, does f have to have a periodic point of some other periodm? A Russian mathematician, A. N Sharkovsky obtained a complete answer to this question. Sharkovsky’s result is elegant, however, his proof is difficult. Recently, P. D. Straffin attempted to give a simple proof of the sufficient part of Sharkovsky’s theorem bymeans of directed graphs. However, his proof contains a gap. In this paper, the authorsfill in the gap in Straffin's work. They also give a proof of the necessary partof thetheorem, which is also based on directed graphs, and thus, obtain a complete simple proof of Sharkovsky’s theorem. © 1981 Pacific Journal of Mathematics.