Spanning tree congestion of the hypercube

Abstract

Let G be a finite connected graph and T be a spanning tree. For any edge e of T , let A e , B e be the components of T ∖ e . The edge congestion of e in T is defined as e c G ( e , T ) = | { u v ∣ u ∈ A e , v ∈ B e } | , and the congestion of T is the maximum of e c G ( e , T ) over all edges of T . Then the spanning tree congestion s ( G ) of G is the minimum of congestion over all spanning trees of G . Hruska [S.W. Hruska, On tree congestion of graphs, Discrete Math. 308 (2008) 1801–1809] conjectured that for the hypercube Q d , s ( Q d ) = 2 d − 1 . We disprove the conjecture and show that s ( Q d ) = Θ ( log 2 d d 2 d ) .