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This paper proves that for every rational number r between 3 and 4, there exists a planar graph G whose circular chromatic number is equal to r . Combining this result with a recent result of Moser, we arrive at the conclusion that every rational number r between 2 and 4 is the circular chromatic...
A graph is totally critical if it is Type 2, connected, and the removal of any edge reduces the total chromatic number. A good characterization of all totally critical graphs is unlikely as Sanchez-Arroyo showed that determining the total chromatic number of a graph is an NP-hard problem. In this...