Recursive Formulae for the Chromatic Polynomials of Complete r-uniform Mixed Interval Hypergraphs

In response to a problem of Voloshin, we find recursive formulae for the chromatic polyno-mials of complete r-uniform interval hypergraphs and cohypergraphs. We also give recursive formulae for the chromatic polynomials of complete 3-uniform and 4-uniform interval bihy-pergraphs and comment on the challenges for general r. Our method is to exploit the uniform and complete structure of these hypergraphs to extend the standard splitting-contraction algorithm to all partitions of a set of order r−1. We then turn the computation of the chromatic polynomials into an exercise in matrix multiplication and comment that this method can be extended to compute the chromatic polynomial of any complete uniform pattern interval hypergraph.