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Programmatically creating functions of many variables

Suppose I have a graph, and I'd like to create a function which checks
whether given assignment of nodes is a proper labeling -- ie, no two
adjacent nodes get the same number. One could do something like the
following, but that gives Part warnings. Is there a preferred

graph = GraphData[{"Grid", {2, 2}}, "AdjacencyMatrix"] // Normal;
f = Function @@ {{x}, And @@ (x[[First[#]]] != x[[Last[#]]] & /@
Position[graph, 1])}

This is actually a problem that I periodically come across. Part
approach causes warnings meanwhile something like Function @@
{Subscript[x,#]&/@Range[n], ...} doesn't work. What are typical ways
of generating multivariate functions automatically?


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