Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài:On Feasible Sets of Mixed Hypergraphs. | On Feasible Sets of Mixed Hypergraphs Daniel Král Department of Applied Mathematics and Institute for Theoretical Computer Science Charles University Malostranske námẽstí 25 118 00 Praha 1 Czech Republic kral@ Submitted Nov 20 2003 Accepted Feb 24 2004 Published Mar 4 2004. MR Subject classifications Primary 05C15 Secondary 05C65 05C85 Keywords coloring of hypergraphs mixed hypergraphs algorithms for coloring Abstract A mixed hypergraph H is a triple V C D where V is the vertex set and C and D are families of subsets of V called C-edges and D-edges. A vertex coloring of H is proper if each C-edge contains two vertices with the same color and each D-edge contains two vertices with different colors. The spectrum of H is a vector ri . rm such that there exist exactly Vi different colorings using exactly i colors rm 1 and there is no coloring using more than m colors. The feasible set of H is the set of all i s such that ri 0. We construct a mixed hypergraph with O Pi log ri vertices whose spectrum is equal to ri . rm for each vector of non-negative integers with r1 0. We further prove that for any fixed finite sets of positive integers Ai c A2 1 ị A2 it is NP-hard to decide whether the feasible set of a given mixed hypergraph is equal to A2 even if it is promised that it is either Ai or A2. This fact has several interesting corollaries . that deciding whether a feasible set of a mixed hypergraph is gap-free is both NP-hard and coNP-hard. 1 Introduction Graph coloring problems are intensively studied both from the theoretical point view and the algorithmic point of view. A hypergraph is a pair V E where E is a family of subsets The author acknowledges partial support by GACR 201 1999 0242 GAUK 158 1999 and KON-TAKT 338 99. Institute for Theoretical Computer Science is supported by Ministry of Education of Czech Republic as project LN00A056. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R19 1 of V of size at least 2. The elements of V are called .
