Triangle contact systems, orthogonal plane partitions and their hit graphs

A triangle contact system S is a geometrical object on the plane consisting of finite number of triangular disks {s(1),...,s(n)} in a triangle Delta bounded by three segments A, B and C such that every vertex of each s(i) coincides with an inner point of either A, B, C or a segment of some s(j). An orthogonal plane partition P is a geometrical object on the plane consisting of a finite number of horizontal segments a(1),..., a(p) and vertical segments b(1),...,b(q) in a rectangle R bounded by four segments A, B, C and D with A horizontal such that every endpoint of each a(i) coincides with an inner point of either B, D or some b(j) and that every endpoint of each b(i) coincides with an inner point of either A, C or some a(j). From S and P we can construct a plane triangulation and a plane quadrangulation according to the adjacency of elements in S and P, respectively. In this paper, we give a survey for triangle contact systems, orthogonal plane partitions and their graphs constructed from them.