Relaxation procedures on graphs

The procedures studied in this paper originate from a problem posed at the International Mathematical Olympiad in 1986. We present several approaches to the IMO problem and its generalizations. In this context we introduce a “signed mean value procedure” and study “relaxation procedures on graphs”. We prove that these processes are always finite, thus confirming a conjecture of Akiyama, Hosono and Urabe [J. Akiyama, K. Hosono, M. Urabe, Some combinatorial problems. Discrete Mathematics 116 (1993) 291–298]. Moreover, we indicate relations to sorting and to an iterative method used in circle packing.