Abstract
For every rational number r such that $3<r<3+1/3$, we construct an infinite family of cubic graphs with circular chromatic index r. Our construction disproves a conjecture of Zhu that there is no bounded increasing infinite sequence in the set of circular chromatic indices of graphs [Topics in Discrete Mathematics, Dedicated to Jarik Neetil on the Occasion of his 60th Birthday, Algorithms Combin. 26, M. Klazar, J. Kratochvíl, M. Loebl, J. Matoušek, R. Thomas, and P. Valtr, eds., Springer, Berlin, 2006, pp. 497–550]. We also describe a sequence of graphs whose circular chromatic indices converge to $3+1/3$ from above.
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