document: The chromatic spectrum of mixed hypergraphs A mixed hypergraph is a triple H = -LRB- X , C , D -RRB- , where X is the vertex set , and each of C , D is a list of subsets of X . A strict k-coloring of H is a surjection c : X -LCB- 1 , ... , k -RCB- such that each member of le has two vertices assigned a common value and each member of D has two vertices assigned distinct values . The feasible set of H is -LCB- k : H has a strict k-coloring -RCB- . Among other results , we prove that a finite set of positive integers is the feasible set of some mixed hypergraph if and only if it omits the number I or is an interval starting with 1 . For the set -LCB- s , t -RCB- with 2 < or = s < or = t - 2 , the smallest realization has 2t - s vertices . When every member of C union D is a single interval in an underlying linear order on the vertices , the feasible set is also a single interval of integers
doc_bio_tags: O B I O B I O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
extractive_keyphrases: chromatic spectrum mixed hypergraphs vertex set positive integers mixed hypergraph
abstractive_keyphrases: strict k-coloring