Abstract
Graph Theory We introduce a variation of chip-firing games on connected graphs. These 'burn-off' games incorporate the loss of energy that may occur in the physical processes that classical chip-firing games have been used to model. For a graph G=(V,E), a configuration of 'chips' on its nodes is a mapping C:V→ℕ. We study the configurations that can arise in the course of iterating a burn-off game. After characterizing the 'relaxed legal' configurations for general graphs, we enumerate the 'legal' ones for complete graphs Kn. The number of relaxed legal configurations on Kn coincides with the number tn+1 of spanning trees of Kn+1. Since our algorithmic, bijective proof of this fact does not invoke Cayley's Formula for tn, our main results yield secondarily a new proof of this formula.
Highlights
Chip-firing games on graphs enjoy a rich literature
This article continues the published account of [20]—initiated in [15]—where we study a variant of chip-firing in which all games have finite length
We present a pair of algorithms that give a one-to-one correspondence between relaxed legal configurations on Kn+1. Let V (Kn+1) (1) Let V (Kn) and spanning trees of Kn+1
Summary
Chip-firing games on graphs enjoy a rich literature This is due to their surprising array of mathematical connections and to their utility in modeling certain kinds of physical systems. For the former, we find ties, e.g., to discrepancy theory [21], the Tutte polynomial [16], critical groups of graphs [5], G-parking functions [3], and stochastic processes [11]. This article continues the published account of [20]—initiated in [15]—where we study a variant of chip-firing in which all games have finite length These ‘burn-off’ games incorporate the loss of energy that may occur in the physical processes that classical chip-firing games have been used to model. For any basics we may have omitted, we point the reader to [8]
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