Runners on the Torn Ballroom Floor

A ballroom floor is an $8 \times 8$ grid of parquet panels, alternating oak and walnut like a checkerboard. Water damage forces the removal of two panels at diagonally opposite corners, leaving $62$. The decorator has $31$ carpet runners, each covering exactly two edge-adjacent panels. Can the $31$ runners exactly cover the $62$ remaining panels, with no overlaps and nothing sticking out? Prove your answer.