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 θ = 2π/k, and connects each point to a nearest neighbor in each cone. It is known for all Yao graphs, with the sole exception of Y5, whether or not they are geometric spanners. In this paper we close this gap by showing that for odd k ≥ 5, the spanning ratio of Y k is at most 1/(1 − 2 sin(3θ/8)), which gives the first constant upper bound for Y5, and is an improvement over the previous bound of 1/(1 − 2 sin(θ/2)) for odd k ≥ 7. We further reduce the upper bound on the spanning ratio for Y5 from 10.9 to 2 + √ 3 ≈ 3.74, which falls slightly below the lower bound of 3.79 established for the spanning ratio of Θ5 (Θ-graphs differ from Yao graphs only in the way they select the closest neighbor in each cone). number of cones. We also give a lower bound of 2.87 on the spanning ratio of Y5. Finally, we revisit the Y6 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 ≤ 6) whose best spanning ratios have been established by complex arguments. Here we reduce the known spanning ratio of Y6 from 17.6 to 5.8, getting closer to the spanning ratio of 2 established for Θ6.

We present improved upper and lower bounds on the spanning ratio of θ-graphs with at least six cones. Given a set of points in the plane, a θ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ = 2π/m, and adds an edge to the 'closest' vertex in each cone. We show that for any integer k ≥ 1, θ-graphs with 4k + 2 cones have a spanning ratio of 1 + 2 sin(θ/2) and we provide a matching lower bound, showing that this spanning ratio tight.Next, we show that for any integer k ≥ 1, θ-graphs with 4k + 4 cones have spanning ratio at most 1 + 2 sin(θ/2)/(cos(θ/2) − sin(θ/2)). We also show that θgraphs with 4k +3 and 4k +5 cones have spanning ratio at most cos(θ/4)/(cos(θ/2)− sin(3θ/4)). This is a significant improvement on all families of θ-graphs for which exact bounds are not known. For example, the spanning ratio of the θ-graph with 7 cones is decreased from at most 7.5625 to at most 3.5132. These spanning proofs also imply improved upper bounds on the competitiveness of the θ-routing algorithm. In particular, we show that the θ-routing algorithm is (1 + 2 sin(θ/2)/(cos(θ/2) − sin(θ/2)))-competitive on θ-graphs with 4k + 4 cones and that this ratio is tight.Finally, we present improved lower bounds on the spanning ratio of these graphs. Using these bounds, we provide a partial order on these families of θ-graphs. In particular, we show that θ-graphs with 4k + 4 cones have spanning ratio at least 1 + 2 tan(θ/2) + 2 tan 2 (θ/2), where θ is 2π/(4k + 4). This is somewhat surprising since, for equal values of k, the spanning ratio of θ-graphs with 4k + 4 cones is greater than that of θ-graphs with 4k + 2 cones, showing that increasing the number of cones can make the spanning ratio worse.

In this paper we study the number of vertex recolorings that an algorithm needs to perform in order to maintain a proper coloring of a graph under insertion and deletion of vertices and edges. We present two algorithms that achieve different trade-offs between the number of recolorings and the number of colors used. For any d > 0, the first algorithm maintains a proper O(CdN 1/d )-coloring while recoloring at most O(d) vertices per update, where C and N are the maximum chromatic number and maximum number of vertices, respectively. The second algorithm reverses the trade-off, maintaining an O(Cd)-coloring with O(dN 1/d ) recolorings per update. The two converge when d = log N , maintaining an O(C log N )-coloring with O(log N ) recolorings per update. We also present a lower bound, showing that any algorithm that maintains a c-coloring of a 2-colorable graph on N vertices must recolor at least Ω(N 2 c(c−1) ) vertices per update, for any constant c ≥ 2.

We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-θ 6 -graph 1 (the half-θ 6 -graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most 5/ √ 3 ≈ 2.887 times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-θ 6 -graph is 2, meaning that even though there always exists a path whose lengths is at most twice the Euclidean distance, we cannot always find such a path when routing locally.Since every triangulation can be embedded in the plane as a half-θ 6 -graph using O(log n) bits per vertex coordinate via Schnyder's embedding scheme (SODA 1990), our result provides a competitive local routing algorithm for every such embedded triangulation. Finally, we show how our routing algorithm can be adapted to provide a routing ratio of 15/ √ 3 ≈ 8.660 on two bounded degree subgraphs of the half-θ 6 -graph.

Let P be a finite set of points in the plane and S a set of non-crossing line segments with endpoints in P . The visibility graph of P with respect to S, denoted Vis(P, S), has vertex set P and an edge for each pair of vertices u, v in P for which no line segment of S properly intersects uv. We show that the constrained half-θ6-graph (which is identical to the constrained Delaunay graph whose empty visible region is an equilateral triangle) is a plane 2-spanner of Vis(P, S). We then show how to construct a plane 6-spanner of Vis(P, S) with maximum degree 6 + c, where c is the maximum number of segments of S incident to a vertex.