Construction of <f>Kn</f>-minor free graphs with given circular chromatic number

For each integer <f>n&ges;5</f> and each rational number <f>r</f> in the interval <f>[2,n−1]</f>, we construct a <f>Kn</f>-minor free graph <f>G</f> with <f>χc(G)=r</f>. This answers a question asked by Zhu (Discrete Mathematics, 229 (1–3) (2001) 371). In case <f>n=5</f>, the constructed graphs are actually planar. Such planar graphs were first constructed in J. Graph Theory 24 (1997) 33 (for <f>∈[2,3]</f>) and in J. Combin. Theory 76 (1999) 170 (for <f>r∈[3,4]</f>). However, our construction and proof are much simpler. [Copyright &y& Elsevier]