Coloring fuzzy circular interval graphs

Given a graph GG with nonnegative node labels ww, a multiset of stable sets  S1,…,Sk⊆V(G)S1,…,Sk⊆V(G) such that each vertex v∈V(G)v∈V(G) is contained in w(v)w(v) many of these stable sets is called a weighted coloring. The weighted coloring number  χw(G)χw(G) is the smallest kk such that there exist stable sets as above.We provide a polynomial time combinatorial algorithm that computes the weighted coloring number and the corresponding colorings for fuzzy circular interval graphs. The algorithm reduces the problem to the case of circular interval graphs, then making use of a coloring algorithm by Gijswijt.We also show that the stable set polytopes of fuzzy circular interval graphs have the integer decomposition property.