Every facial boundary walk has length at least five. Bridge excursions can repeat an edge, but deleting those excursions leaves a boundary containing a cycle in a graph; the finite girth assumption ensures the graph is not a tree. Thus the shortest-cycle condition still gives the lower bound five even when bridges occur. Summing boundary lengths counts each edge twice, so . Using Euler formula for a connected planar graph,
(1)
Hence
(2)

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