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:Branch-width, parse trees, and monadic second-order logic for matroids, 20th Annual Symposium on Theoretical Aspects of Com- puter Science (STACS), LNCS vol
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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.