On cross-intersecting families of independent sets in graphs

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 is an element of[k] with i not equal j, A is an element of A(i) and B is an element of A(j) implies A boolean AND B not equal phi. We consider a theorem of Hilton which gives a best possible upper bound on the sum of the cardinalities of uniform cross-intersecting families. 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 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.