Covering and coloring problems for relatives of intervals

Abstract The following generalizations and relatives of interval families are studied in the paper: arcs of a circle, multiple intervals, chords of a circle, d -dimensional boxes and multiple boxes. For these families we survey results and problems concerning the dependence of the transversal number on the packing number and the dependence of the coloring number on the clique number. The intersection graphs of the underlying set systems are circular arc graphs, multiple interval graphs, circle graphs (called also overlap graphs), box graphs and multiple box graphs. Thus most of our problems and results concern the relation between the clique-cover number and stability number (ϑ and α), or between the chromatic number and clique number (χ and ω) of these graphs.