  Our current understanding of interstellar carbon fractionation hinges on the
interpretation of astrochemical kinetic models. Yet, the various reactions
included carry large uncertainties in their (estimated) rate coefficients,
notably those involving C with C$_{2}$. In this work, we provide theoretical
thermal rate coefficients as a function of the temperature for all possible
gas-phase isotope-exchange reactions of
C+C$_{2}(X^{1}\Sigma_{g}^{+},a^{3}\Pi_{u})$. For this, we employ the
quasi-classical trajectory method, with the previously obtained potential
energy surfaces of C$_{3}$ dictating the forces between the colliding partners.
The calculated rate coefficients show a positive temperature dependence and are
markedly different from previous theoretical estimates. While the forward
reactions are fast and inherently exothermic owing to the lower zero-point
energy content of the products, the reverse processes have temperature
thresholds. For each reaction considered, analytic three-parameter
Arrhenius-Kooij formulas are provided that readily interpolate/extrapolate the
associated forward and backward rates. These forms can further be introduced in
astrochemical networks. Apart from the proper kinetic attributes, we also
provide equilibrium constants for these processes, thence confirming their
prominence in the overall C fractionation chemistry. In this respect, the
$^{13}$C + $^{12}$C$_{2}(X^{1}{\Sigma}^{+}_{g})$ and $^{13}$C +
$^{12}$C$_{2}(a^{3}{\Pi}_{u})$ reactions are found to be particularly
conspicuous, notably at the typical temperatures of dense molecular clouds. For
these reactions and considering both equilibrium and time-dependent chemistry,
theoretical $^{12}$C/$^{13}$C ratios as a function of the gas kinetic
temperature are also derived and shown to be consistent with available model
chemistry and observational data on C$_{2}$
