  Let A_1,...,A_k be a collection of families of subsets of an n-element set.
We say that this collection is cross-intersecting if for any i,j in [k] with i
not equal to j, A in A_i and B in A_j implies that the intersection of A and B
is nonempty. We consider a theorem of Hilton which gives a best possible upper
bound on the sum of the cardinalities of uniform cross-intersecting
subfamilies. We formulate a graph-theoretic analogue of Hilton's
cross-intersection theorem, similar to the one developed by Holroyd, Spencer
and Talbot for the Erdos-Ko-Rado theorem. In particular we build on a result of
Borg and Leader for signed sets and prove a theorem for uniform
cross-intersecting subfamilies of independent vertex subsets of a disjoint
union of complete graphs. We proceed to obtain a result for a much larger class
of graphs, namely chordal graphs and propose a conjecture for all graphs. We
end by proving this conjecture for the cycle on n vertices.
