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Edge-Simple Circuits Through 10 Ordered Vertices in Square Grids

David Coudert 1, * Frédéric Giroire 1 Ignasi Sau Valls 1, 2
* Corresponding author
1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : A circuit in a simple undirected graph G = (V , E) is a sequence of vertices {v1 , v2 , . . . , vk+1 } such that v1 = vk+1 and {vi , vi+i } ∈ E for i = 1, . . . , k. A circuit C is said to be edge-simple if no edge of G is used twice in C . In this article we study the following problem: which is the largest integer k such that, given any subset of k ordered vertices of an infinite square grid, there exists an edge-simple circuit visiting the k vertices in the prescribed order? We prove that k = 10. To this end, we first provide a counterexample implying that k < 11. To show that k ≥ 10, we introduce a methodology, based on the notion of core graph, to reduce drastically the number of possible vertex configurations, and then we test each one of the resulting configurations with an ILP solver.
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Submitted on : Saturday, October 31, 2009 - 3:12:59 PM
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David Coudert, Frédéric Giroire, Ignasi Sau Valls. Edge-Simple Circuits Through 10 Ordered Vertices in Square Grids. International Workshop on Combinatorial Algorithms -- IWOCA, Jun 2009, Hradec nad Moravicì, Czech Republic. ⟨inria-00429146⟩

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