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Eulerian cycle in a grid graph

Source: 2022 China TST, Test 2, P1

March 28, 2022
graph theorygridcombinatorics

Problem Statement

Find all pairs of positive integers (m,n)(m, n), such that in a m×nm \times n table (with m+1m+1 horizontal lines and n+1n+1 vertical lines), a diagonal can be drawn in some unit squares (some unit squares may have no diagonals drawn, but two diagonals cannot be both drawn in a unit square), so that the obtained graph has an Eulerian cycle.