Abstract
Disproving a conjecture of Máčajová, Raspaud and Škoviera, Kardoš and Narboni constructed a non-4-colourable simple signed planar graph, or equivalently, a simple signed planar graph with circular chromatic number greater than 4. Naserasr, Wang, and Zhu improved this result by constructing a simple signed planar graph with circular chromatic number 14/3. This paper further strengthens this result by constructing, for each rational 4<p/q≤14/3, a simple signed planar graph with circular chromatic number p/q. Together with some earlier results of Moser and Zhu, this implies that every rational p/q∈[2,14/3] is the circular chromatic number of a simple signed planar graph.
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