A disproof of the conjecture about boundary H -points of H -triangles

An H-triangle is a triangle with corners in the set of vertices of a tiling of \(\mathbb {R}^{2}\) by regular hexagons of unit edge. It is known that any H-triangle with k interior H-points can have at most \(3k+7\) boundary H-points and can not have \(3k+6\) boundary H-points. In this note we prove that any H-triangle with exactly k interior H-points can not have 3k boundary H-points for \(k\ge 5\).