Any spanning tree of the n x n grid has fundamental cycles with average length Omega(log n), and the bound is tight.
Hlinˇ en´ y:On matroid properties definable in the MSO logic, 27th International Symposium on Mathematical Foundations of Computer Science, MFCS (2003), 470–479
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Fundamental cycles in grid graphs
Any spanning tree of the n x n grid has fundamental cycles with average length Omega(log n), and the bound is tight.