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The Circular Chromatic Number of SeriesParallel Graphs
"... In this paper, we consider the circular chromatic number c (G) of seriesparallel graphs G. It is well known that seriesparallel graphs have chromatic number at most 3. Hence their circular chromatic number is also at most 3. If a seriesparallel graph G contains a triangle, then both the chro ..."
Abstract

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In this paper, we consider the circular chromatic number c (G) of seriesparallel graphs G. It is well known that seriesparallel graphs have chromatic number at most 3. Hence their circular chromatic number is also at most 3. If a seriesparallel 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 seriesparallel graph G has girth at least 2b(3k \Gamma 1)=2c, then c (G) 4k=(2k \Gamma 1). The special case k = 2 of this result implies that a triangle free seriesparallel graph G has circular chromatic number at most 8=3. Therefore the circular chromatic number of a seriesparallel 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. 1 1