Revisiting Tuckerʹs Algorithm to Color Circular-Arc Graphs

The circular arc coloring problem consists of finding a minimal coloring of a circular arc family F such that no two intersecting arcs share a color. Let l be the minimum number of circular arcs in F that are needed to cover the circle such that the intersection graph induced by these arcs is a cycle. Tucker shows in [8], that if l ≥ 4, then [3/2L] colors suffice to color F, where L denotes the load of F. We extend Tuckerʹs result by showing that if l ≥ 4, then [(l-1/l-2) L] + 1 colors suffice to color F.