On rectangle intersection and overlap graphs

Let R be a family of isooriented rectangles in the plane. A graph G=(V, E) is called a rectangle intersection (respectively, overlap) graph for R if there is a one-to-one correspondence between V and R such that two vertices in V are adjacent to each other if and only if the corresponding rectangles in R intersect (respectively, overlap) each other. We first prove that the maximum independent set problem is NP-hard even for both cubic planar rectangle intersection and cubic planar rectangle overlap graphs. We then show the NP-completeness of the vertex coloring problem with three colors for both planar rectangle intersection and planar rectangle overlap graphs even when the degree of every vertex is limited to four. These NP-hardness results are obtained for the tightest degree constraint cases