For bunkbed graphs with reflection-symmetric capacities, same-side max flow dominates cross-side max flow, and for K_n x K_2, self-avoiding walks to the mirrored vertex outnumber same-side walks for n=3,4,5 and all sufficiently large n.
On a conjecture of Bollob´ as and Brigh twell concerning random walks on product graphs
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Maximum flow and self-avoiding walk on bunkbed graphs
For bunkbed graphs with reflection-symmetric capacities, same-side max flow dominates cross-side max flow, and for K_n x K_2, self-avoiding walks to the mirrored vertex outnumber same-side walks for n=3,4,5 and all sufficiently large n.