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  "path": "/abs/2603.09048v1",
  "publishedAt": "2026-03-11T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Prosenjit Bose",
    "Jean-Lou De Carufel",
    "Darryl Hill",
    "John Stuart"
  ],
  "textContent": "**Authors:** Prosenjit Bose, Jean-Lou De Carufel, Darryl Hill, John Stuart\n\nGiven a finite set $P\\subset\\mathbb{R}^2$, the directed Theta-6 graph, denoted $\\vecΘ_6(P)$, is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The $\\vecΘ_6(P)$-graph is defined as follows: the plane around each point $u\\in P$ is partitioned into $6$ equiangular cones with apex $u$, and in each cone, $u$ is joined to the point whose projection on the bisector of the cone is closest. Equivalently, the $\\vecΘ_6(P)$-graph contains an edge from $u$ to $v$ exactly when the interior of $\\nabla_u^v$ is disjoint from $P$, where $\\nabla_u^v$ is the unique equilateral triangle containing $u$ on a corner, $v$ on the opposite side, and whose sides are parallel to the cone boundaries. It was previously shown that the spanning ratio of the $\\vecΘ_6(P)$-graph is between $4$ and $7$ in the worst case (Akitaya, Biniaz, and Bose \\emph{Comput. Geom.}, 105-106:101881, 2022). We close this gap by showing a tight spanning ratio of 5. This is the first tight bound proven for the spanning ratio of any $\\vecΘ_k(P)$-graph. Our lower bound models a long path by mapping it to a converging series. Our upper bound proof uses techniques novel to the area of spanners. We use linear programming to prove that among several candidate paths, there exists a path satisfying our bound.",
  "title": "The Spanning Ratio of the Directed $Θ_6$-Graph is 5"
}