On the chromatic number of multiple interval graphs and overlap graphs

Let x(G) and w(G) denote the chromatic number and clique number (maximum size of a clique) of a graph G. To avoid trivial cases, we always assume that w (G);?: 2. It is well known that interval graphs are perfect, in particular x( G)= w (G) for every interval graph G. In this paper we study the closeness of x and w for two well-known non-perfect relatives of interval graphs: multiple interval graphs and overlap graphs. Multiple interval graphs are the intersection graphs of sets Ab A 2 , ••• , An such that for all i, 1 ~ i ~ n, Ai is the union of closed intervals of the real line. If for all i, 1 ~ i ~ n, Ai is the union of t closed intervals then we speak about t-interval graphs. Multiple interval graphs were introduced by Harary and Trotter in [9]. Relations betwen the packing number and transversal number of multiple intervals were studied in [6]. Obviously, 1-interval graphs are exactly the interval graphs. It is easy to see that 2-interval graphs (or double interval graphs) include another distinguished family of graphs, the circular arc graphs. Circular-arc graphs are the intersection graphs of closed arcs of a circle. It is straightforward that x ~ 2w holds for circular-arc graphs. A conjecture of Tucker states that x ~ L~w J for circular-arc graphs [10]. We shall prove that x~2t(w -1) holds fort-interval graphs (Theorem 1). Overlap graphs are graphs whose vertices can be ·put into one-to-one correspondence with a collection of intervals on a line in such a way that two vertices are adjacent if and only if the corresponding intervals intersect but neither contains the other. Overlap graphs can be equivalently defined as intersection graphs of chords of a circle (see [5, Ch. 11.3]). We shall prove that x ~ 2ww 2 (w -1) holds for overlap graphs (Theorem 2.).