The circular chromatic number is a new concept in graph theory. More and more people working on graph coloring and flow problems are getting interested in this concept.

This paper reviews current research results on this topic, and it also contains a list of open problems. It puts the research results in a unified framework, and it is a convenient source of information on this subject.

Circular coloring of a graph colors the vertices of the graph by points of a circle in such a way that adjacent vertices are colored by points that are far apart in the circle. It is a variation of the traditional vertex coloring of graphs, in which the vertices are colored by integers, and colors of adjacent vertices are distinct. The circular chromatic number is a refinement of the chromatic number of a graph, and reveals more information about the structure of the graph. Both coloring and circular coloring find applications in resource distribution problems. However, when applied to resource distribution problems in which the consumers use the resource in a periodic way, the circular coloring gives more efficient distributions. Such resource distribution problems are very common in computer science. Current research on circular chromatic number has been concentrated on the relation between the circular chromatic number and other graph parameters. It is hoped that the concept will find more applications in computer science and other areas.

Professor Xuding Zhu
Department of Applied Mathematics
National Sun Yat-sen University
Kaohsiung, Taiwan