The interval number of a planar graph: Three intervals suffice

Abstract Suppose each vertex of a graph G is assigned a subset of the real line consisting of at most t closed intervals. This assignment is called a t -interval representation of G when vertex v is adjacent to vertex w if and only if some interval for v intersects some interval for w . The interval number i ( G ) of a graph G is the smallest number t such that G has a t -interval representation. It is proved that i ( G ) ≤ 3 whenever G is planar and that this bound is the best possible. The related concepts of displayed interval number and depth- r interval number are discussed and their maximum values for certain classes of planar graphs are found.