  We develop further the new versions of quantum chromatic numbers of graphs
introduced by the first and fourth authors. We prove that the problem of
computation of the commuting quantum chromatic number of a graph is solvable by
an SDP algorithm and describe an hierarchy of variants of the commuting quantum
chromatic number which converge to it. We introduce the tracial rank of a
graph, a parameter that gives a lower bound for the commuting quantum chromatic
number and parallels the projective rank, and prove that it is multiplicative.
We describe the tracial rank, the projective rank and the fractional chromatic
numbers in a unified manner that clarifies their connection with the commuting
quantum chromatic number, the quantum chromatic number and the classical
chromatic number, respectively. Finally, we present a new SDP algorithm that
yields a parameter larger than the Lov\'asz number and is yet a lower bound for
the tracial rank of the graph. We determine the precise value of the tracial
rank of an odd cycle.
