Circuit visiting 10 ordered vertices in infinite grids

David Coudert (1) , Frédéric Giroire (1) , Ignasi Sau (1, 2)

1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
2 Applied Mathematics IV Department

A circuit in a simple undirected graph G=(V,E) is a sequence of vertices {v_1,v_2,...,v_{k+1}} such that v_1=v_{k+1} and {v_i,v_{i+1}} \in 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.

Mots clés: Connectivity; simple circuit; grid

Domaines: Mathématique discrète [cs.DM]

David Coudert, Frédéric Giroire, Ignasi Sau. Circuit visiting 10 ordered vertices in infinite grids. [Research Report] RR-6910, INRIA. 2009. ⟨inria-00378586⟩