Relaxation procedures on graphs

Abstract

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.

Chip-firing games, potential theory on graphs, and spanning trees

“…We refer the reader to[49,47,45,2,40] for some discussions, solutions, and generalizations of this problem.…”

Weighted Coxeter graphs and generalized geometric representations of Coxeter groups

“…Prove that this game must always terminate. Several solutions to this problem can be found in [14]. Eriksson and Björner found deep connections between the numbers game and Coxeter groups: taking the positive weights k ij satisfying k ij k ji = 4 cos 2 (π/m ij ) where m ij is the minimal number such that for Coxeter generators s i and s j there is the relation (s i s j ) m ij = 1 and k ij k ji 4 when the element s i s j has infinite order, the numbers game becomes a combinatorial model of the Coxeter group, where group elements correspond to positions and reduced decompositions correspond to legal play sequences, see Chapter 4 in [3] and [7] for details.…”