Circuits in Graphs through a prescribed Set of Ordered vertices

A circuit in a simple undirected graphG = (V,E) is a sequence of vertices {v1, v2, . . . , vk+1} such that v1 = vk+1 and {vi, vi+1} ∈ E for i = 1, . . . , k. A circuit C is said to be edgesimple 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 a graph G, there exists an edge-simple circuit visiting the k vertices in the prescribed order? We first study the case when G has maximum degree at most 3, establishing the value of k for several subcases, such as when G is planar or 3-vertex-connected. Our main result is that k = 10 in infinite square grids. To prove this, we introduce a methodology based on the notion of core graph, in order to reduce the number of possible vertex configurations, and then we test each one of the resulting configurations with an Integer Linear Program (ILP) solver.