A mixed hypergraph is a triple H=(V,C,D), where V is
a set of vertices, C and D are sets of hyperedges. A
vertex-coloring of H is proper if C-edges are not totally multicolored and
D-edges are not monochromatic. The feasible set S(H) of H is the set of
all integers, s, such that H has a proper coloring with s colors.
Bujt\'as and Tuza [Graphs and Combinatorics 24 (2008), 1--12] gave a
characterization of feasible sets for mixed hypergraphs with all C- and
D-edges of the same size r, rβ₯3.
In this note, we give a short proof of a complete characterization of all
possible feasible sets for mixed hypergraphs with all C-edges of size β
and all D-edges of size m, where β,mβ₯2. Moreover, we show that
for every sequence (r(s))s=βnβ, nβ₯β, of natural numbers there
exists such a hypergraph with exactly r(s) proper colorings using s colors,
s=β,β¦,n, and no proper coloring with more than n colors. Choosing
β=m=r this answers a question of Bujt\'as and Tuza, and generalizes
their result with a shorter proof.Comment: 9 pages, 5 figure