Chromatic polynomials and representations of the symmetric group Original Research Article

The chromatic polynomials considered in this paper are associated with graphs constructed in the following way. Take n copies of a complete graph Kb and, for i=1,2,…,n, join each vertex in the ith copy to the same vertex in the (i+1)th copy, taking n+1=1 by convention. Previous calculations for b=2 and b=3 suggest that the chromatic polynomial contains terms that occur in ‘levels’. In the present paper the levels are explained by using a version of the sieve principle, and it is shown that the terms at level ℓ correspond to the irreducible representations of the symmetric group Symℓ. In the case of the two linear representations the terms can be calculated explicitly by methods based on the theory of distance-regular graphs. For the nonlinear representations the calculations are more complicated. An illustration is given in Section 10, where the complete chromatic polynomial for the case b=4 is obtained.