On cross-intersecting families of independent sets in graphs

Let A1, . . . ,Ak 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 ∈ [k] with i = j, A ∈ Ai and B ∈ Aj implies A ∩ B = ∅. 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 graphtheoretic analogue of Hilton’s cross-intersection theorem, similar to the one developed by Holroyd, Spencer and Talbot for the Erdős-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.