The bunkbed conjecture on the complete graph

Abstract The bunkbed conjecture was first posed by Kasteleyn. If G = ( V , E ) is a finite graph and H some subset of V , then the bunkbed of the pair ( G , H ) is the graph G × { 1 , 2 } plus | H | extra edges to connect for every v ∈ H the vertices ( v , 1 ) and ( v , 2 ) . The conjecture asserts that ( v , 1 ) is more likely to connect with ( w , 1 ) than with ( w , 2 ) in the independent bond percolation model for any v , w ∈ V . This is intuitive because ( v , 1 ) is in some sense closer to ( w , 1 ) than it is to ( w , 2 ) . The conjecture has however resisted several attempts of proof. This paper settles the conjecture in the case of a constant percolation parameter and G the complete graph.