The discussion focuses on clarifying a proof involving a procedure where walkers traverse edges in a specific order, starting from the shortest to the longest. It emphasizes that at each stage, two walkers exchange positions along an edge, contributing to the total distance traveled. The total number of edges traversed is noted as 2m, where m is the number of edges, and with n walkers, the average distance traveled per walker is calculated as 2m/n. This leads to the conclusion that at least one walker must have traveled a distance of 2m/n or more edges. The proof is acknowledged for its elegance and clarity, enhancing understanding of the concept.