Adventures in Graph Theory by W. David Joyner & Caroline Grant Melles

Author:W. David Joyner & Caroline Grant Melles
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Corollary 4.3.23.

A q-stable configuration s is q-critical, if and only if there is an ordering of the vertices of , such that the corresponding sequence of chip-firings is q-legal and takes s to s, i.e., if and only if there exists a q-legal complete q-pointed sequence of chip-firings.

Proof.

The result follows immediately from Proposition 4.3.22, since any q-legal complete q-pointed sequence of chip-firings is necessarily q-recurrent.

Exercise 4.8.

Let be the diamond graph of Figure 4.1. Show that the configuration is q-critical by showing that the sequence of vertex-firings is q-legal.

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