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Walks on Borders of Polygons

We give a graph theoretical proof for the existence of a walk connecting left and right sides of a rectangle when it is partitioned into (finitely many) rectangles colored in a specific way, and the walk is aloud to use only borders between rectangles of different color. The problem was originally introduced by J.R. Isbell in the proof of his Zig-Zag Theorem. We prove the existence of such a walk also when the problem is generalized other partitions of the rectangle, such as partitions to arbitrary polygons.