Programmatically creating multivariate functions in Mathematica

Question

This is a split from discussion on earlier question.

Suppose I need to define a function f which checks if given labeling of a graph is a proper coloring. In other words

Answer 1

I'm still puzzled by the difficulties that you seem to be having. Here's a function which checks that no 2 consecutive elements in a list are identical:

f[l_List] := Length[Split[l]] == Length[l]

No trouble with Part, no error messages for the simple examples I've tried so far including OP's 'test' cases. I also contend that this is more concise and more readable than either of the other approaches seen so far.