King independence on triangle boards (Q1810633)

The authors consider game boards \(B_n\) that are sets of triangles of the Euclidean triangle tessellation such that all triangles having at least one point in common with \(B_{n-2}\) are added to obtain \(B_n\), where \(B_1\) is a single triangle and \(B_2\) consists of six triangles meeting at one point. The king's graph for such a game board has the triangles of \(B_n\) as vertices and all possible moves of a king determine the edges. Two different types of moves defined lead to two separate cases to consider. The independence number of such graphs are determined completely for one type of move and partially for the other.