Improved Bounds on the Stretch Factor of $Y_4$

We establish an upper bound of 4.94 on the stretch factor of the Yao graph $Y_4^\infty$ defined in the $L_\infty$-metric, improving upon the best previously known upper bound of 6.31. We also establish an upper bound of 54.62 on the stretch factor of the Yao graph $Y_4$ defined in the Euclidean metric, improving upon the best previously known upper bound of 662.16.