  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.
