If a graph is 4-list-colorable, then it is easy to see that it has exponentially many 5-list colorings. This is the first sentence in [Exponentially many 5-list-colorings of planar graphs,JCTB,97 (2007) 571–583]

I do not understand why a graph must have exponentially many 5-list colorings when it is 4-choosable.

Could any one explain the sentence above. Thanks


Take $5$-lists at every vertex. There are $5^n$ ways to reduce them to $4$-lists. Each of these ways allows a coloring, so you have $5^n$ pairs of a $4$-list and a coloring.

How many of these colorings can be identical? Given a coloring, $4^n$ of the $4$-lists allow the same coloring, as you are not allowed to delete the particular color from the list. Double counting the pairs, you get that there are at least $\left(\frac54\right)^n$ colorings total.


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