Computations of fractional chromatic numbers this week tell me that for Fullerene Graphs the value is $5/2$. I have computed $100$ of these or more. Is there any theorem that would say this? Any information on formulas for fractional chromatic numbers of families of graphs would be welcome. I am aware that the Kneser graphs $K(a,b)$ have fractional chromatic number $a/b$. For the definition of a Fullerene graph see this MathWorld link.
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Any fullerene contains pentagons. Each of the 5 vertices on a pentagon touches 2 others. At most 2 vertices on any pentagon can share a colour since adjacent vertices have different colours. Therefore the chromatic number of a pentagon is 5/2.
If some of the vertices of a pentagon are coloured then, if for example a vertex is red and blue, the 2 opposite vertices must be coloured so that one is red and the other is blue. A fullerene can be coloured with 5/2 colours by first colouring a pentagon, then all the adjacent pentagons, then any adjacent to the adjacent pentagons... then any adjacent hexagons, then another pentagon and its adjacent pentagons, then adjacent hexagons etc.