We establish an upper bound of 13 + 8 √ 2 4.931 on the stretch factor of the Yao graph Y ∞ 4 defined in the L∞-metric, improving upon the best previously known upper bound of 6.31. We also establish an upper bound of (11 + 7 √ 2) 4 + 2 √ 2 54.62 on the stretch factor of the Yao graph Y4 defined in the L2-metric, improving upon the best previously known upper bound of 662.16.