Interval Representations of Cliques and of Subset Intersection Graphs a

In this note, we consider the use of intervals to represent two classes of highly symmetric graphs, in fact with n!-fold symmetry. The first are the complete graphs. The second are the graphs whose vertices correspond to the subsets of an n-set, with vertices adjacent if and only if the corresponding sets intersect. The symmetry is important in each discussion. First we define the parameters to be studied. Consider representing a graph by assigning each vertex v a subset f ( v ) of the real line, such that vertices are adjacent if and only if the corresponding subsets intersect. If each vertex is assigned a set consisting of at most t intervals, we have a t-interval representation. The interval number i(G) of a graph G is the minimum t such that G has a t-interval representation. The graphs with interval number 1 are called interval graphs and have been thoroughly studied and characterized. If each point of the line appears in sets assigned to at most r vertices of G , the representation has depth r. The depth r interval number i iG) is the minimum t such that G has a t-interval representation of depth r. Letting A(G) denote the maximum vertex degree in G and w(G) the maximum clique size, note that i(G) = i,(G) 5 . . . I i2(G) I A(G), so all these parameters are well defined. In fact, Griggs and West [l] showed i2(G) I; r(A(G) + 1)/21. In the next section, we show