The circular chromatic number of series-parallel graphs of large odd girth

In this paper, we consider the circular chromatic number c (G) of series-parallel graphs G. It is well known that series-parallel graphs have chromatic number at most 3. Hence their circular chromatic number is also at most 3. If a series-parallel graph G contains a triangle , then both the chromatic number and the circular chromatic number of G are indeed equal to 3. We shall show that if a series-parallel graph G has girth at least 2b(3k ? 1)=2c, then c (G) 4k=(2k ? 1). The special case k = 2 of this result implies that a triangle free series-parallel graph G has circular chromatic number at most 8=3. Therefore the circular chromatic number of a series-parallel graph (and of a K 4-minor free graph) is either 3 or at most 8=3. This is in sharp contrast to recent results of Moser 4] and Zhu 9], which imply that the circular chromatic number of K 5-minor free graphs are precisely all rational numbers between 2 and 4. We shall also construct examples to demonstrate the sharpness of the bound given in this paper.