For a set of points in the plane and a fixed integer \(k > 0\), the Yao graph \(Y_k\) partitions the space around each point into \(k\) equiangular cones of angle \(\theta=2\pi/k\), and connects each point to a nearest neighbor in each cone. It is known for all Yao graphs, with the sole exception of \(Y_5\), whether or not they are geometric spanners. In this talk, we close this gap by showing that for odd \(k \geq 5\), the spanning ratio of \(Y_k\) is at most \(1/(1-2\sin(3\theta/8))\), which gives the first constant upper bound for \(Y_5\), and is an improvement over the previous bound of \(1/(1-2\sin(\theta/2))\) for odd \(k \geq 7\).

We further reduce the upper bound on the spanning ratio for \(Y_5\) from \(10.9\) to \(2+\sqrt{3} = 3.74\), which falls slightly below the lower bound of \(3.79\) established for the spanning ratio of \(\Theta_5\) (\(\Theta\)-graphs differ from Yao graphs only in the way they select the closest neighbor in each cone). This is the first such separation between a Yao and \(\Theta\)-graph with the same number of cones. We also give a lower bound of \(2.87\) on the spanning ratio of \(Y_5\).

In addition, we revisit the \(Y_6\) graph, which plays a particularly important role as the transition between the graphs (\(k > 6\)) for which simple inductive proofs are known, and the graphs (\(k \le 6\)) whose best spanning ratios have been established by complex arguments. Here we reduce the known spanning ratio of \(Y_6\) from \(17.6\) to \(5.8\), getting closer to the spanning ratio of 2 established for \(\Theta_6\).