# What is the definition of a discharge rule?

This question is in the reverse direction of a common MO question. Instead of being faced with a formal definition and asking for some intuition for the definition, I have a concept with I understand intuitively but where I do not know a definition.

I have been in interested in the proofs of the 4-colour theorem which uses the notion of a "discharge rule". This evening I was discussing discharge rules with a speaker who was not familiar with the proofs of the 4-colour theorem and was embarrassed that I could not give a definition. I have seen examples of discharge rules and I feel I would know one if I saw one. However I have no recollection of reading a definition. Did I overlook a definition in one of the papers? and if not, could someone complete the following sentence:

Definition A discharge rule is ...

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Is a rule with which the charge of a vertex/face is redistributed to other vertices/faces? –  Jernej May 9 '12 at 19:13
@Jernej Yes, it is, and that would be a good starting point in a face to face discussion as it describes how discharge rules are used. However the definition needs to be more restrictive. –  Bruce Westbury May 9 '12 at 20:13
Well, first you might want to define rigorously what is a 'charged graph', and then the set $CG$ of charged graphs comes with a map to the set $G$ of graphs. There will probably be interesting functions from $CG$ to the integers (or poss. rationals) giving invariants of a charged graph. Or maybe they will be maps to $\bigoplus \mathbb{Z}$. A discharge rule (hand-waving alert!) could be an endomorphism of $CG$ such that invariants are preserved, or decreased, or something. –  David Roberts May 10 '12 at 4:50
@David. Yes, we need to define a charged graph but I don't see any issues there. –  Bruce Westbury May 10 '12 at 9:51