The circular arc coloring problem consists of finding a minimum 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. Tucker shows in [SIAM J. Appl. Math., 29 (1975), pp. 493--502], that if $l \geq 4$, then $\lfloor \frac{3}{2}L \rfloor$ colors suffice to color F, where L denotes the load of F. We extend Tucker's result by showing that if $l \geq 5$, then $\lceil (\frac{l-1}{l-2} ) L \rceil$ colors suffice to color F, and this upper bound is tight.
