 Brooks's theorem states that for non-complete graphs that are not odd cycles, this quantity is at most equal to δ, the maximum vertex degree
Godsil and Royle showed that the fractional version of this quantity is equal to the fractional clique number, and that it is less than or equal to this quantity. Erdös proved that there exist graphs with arbitrarily large girth and this quantity. Brooks's theorem states that for non-complete graphs that are not odd cycles, this quantity is at most equal to δ, the maximum vertex degree. This quantity is equal to n for a   complete graph with n vertices. This quantity is usually denoted with the letter χ [chi], and for planar graphs this quantity is at most four. For ten points, identify this quantity from graph theory, which gives the minimum number of colors required to color a given graph.
Godsil and Royle showed that the fractional version of this quantity is equal to the fractional clique number, and that it is less than or equal to this quantity.
chromatic number [prompt on descriptions like  the number of colors required to color a graph  or similar before the last clue is read]