- Email: [email protected]
Mixed colorings of hypergraphs
Electronic Notes in Discrete Mathematics 24 (2006) 273–275 www.elsevier.com/locate/endm
Mixed colorings of hypergraphs Csilla Bujt´as 1,2 Department of Computer Science Pannon University Veszpr´em, Hungary
Zsolt Tuza 1,3 Computer and Automation Institute Hungarian Academy of Sciences Budapest, Hungary and Department of Computer Science Pannon University Veszpr´em, Hungary
Abstract We generalize the concept of the colorings of mixed hypergraphs to ‘mixed colorings’ of hypergraphs. The idea is to put lower and upper bounds on the number of colors assigned to the vertices in each edge. Some earlier results can be extended for our model, too, while some others — e.g., concerning time complexity — change in a significant way. Keywords: Hypergraph, vertex coloring, chromatic polynomial, feasible set, interval hypergraph, hypertree, mixed hypergraph, unique colorability, gap in the chromatic spectrum, perfect hypergraph.
1571-0653/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2006.06.026
274
C. Bujtás, Z. Tuza / Electronic Notes in Discrete Mathematics 24 (2006) 273–275
Summary of talk We investigate vertex colorings of hypergraphs such that, for each edge, a lower and an upper bound are given on the number of colors occurring on the vertices of the edge. This generalizes the concept of ‘mixed hypergraph’ introduced by Voloshin [5], as ‘D-edge’ means lower bound 2, while ‘C-edge’ assumes that the lower bound is 1 and the upper bound is one smaller than the cardinality of the edge; cf. [6] as a comprehensive source of information. The ‘bi-edges’ occurring in the theory of mixed hypergraphs are represented in the frame of ‘mixed coloring’ in a natural, unified way. Our model also extends the one considered by Drgas-Burchardt and Lazuka [3] where just a lower bound not exceeding the edge size is given on the number of colors in each edge. In the talk we give a brief summary of results presented in two recent manuscripts, the first one [1] discussing general observations while the second one [2] concentrating on the major role of hypertrees (hypergraphs whose edges are vertex sets of subtrees of a tree). We mostly consider problems concerning the chromatic spectrum, which is the sequence whose k-th element is the number of allowed colorings with precisely k colors. In particular, the lower and upper chromatic number — the minimum and maximum number of colors in a coloring — are of great interest. We also prove complexity results much stronger than those valid for mixed hypergraphs, e.g. concerning unique colorability. A related but somewhat different generalization of mixed hypergraphs is introduced, too, by imposing conditions on the number of occurrences of each color in any edge. Similarly to the model described above, it is allowed that for different edges the coloring restrictions be different. Some facts remain valid for ‘pattern hypergraphs’ as well, a structure class introduced and studied recently by Dvoˇr´ak et al. in [4].
1
Research supported in part by the Hungarian Scientific Research Fund, OTKA grant T-049613. 2 Email: [email protected] 3 Email: [email protected]
C. Bujtás, Z. Tuza / Electronic Notes in Discrete Mathematics 24 (2006) 273–275
275
References [1] Bujt´as, Cs., and Zs. Tuza, Hypergraph mixed coloring, I: General results, preprint, 2006. [2] Bujt´as, Cs., and Zs. Tuza, Hypergraph mixed coloring, II: Interval hypergraphs and hypertrees, preprint, 2006. [3] Drgas-Burchardt, E., and E. Lazuka, On chromatic polynomials of hypergraphs, preprint, 2005. [4] Dvoˇr´ak, Z., J. K´ara, D. Kr´al’, and O. Pangr´ ac, Pattern hypergraphs, preprint, 2004. [5] Voloshin, V. I., On the upper chromatic number of a hypergraph, Australasian J. Combin. 11 (1995), 25–45. [6] Voloshin, V. I., “Coloring Mixed Hypergraphs – Theory, Algorithms and Applications,” Fields Institute Monographs 17, Amer. Math. Soc., Providence, R.I., 2002.
Mixed colorings of hypergraphs Csilla Bujt´as 1,2 Department of Computer Science Pannon University Veszpr´em, Hungary
Zsolt Tuza 1,3 Computer and Automation Institute Hungarian Academy of Sciences Budapest, Hungary and Department of Computer Science Pannon University Veszpr´em, Hungary
Abstract We generalize the concept of the colorings of mixed hypergraphs to ‘mixed colorings’ of hypergraphs. The idea is to put lower and upper bounds on the number of colors assigned to the vertices in each edge. Some earlier results can be extended for our model, too, while some others — e.g., concerning time complexity — change in a significant way. Keywords: Hypergraph, vertex coloring, chromatic polynomial, feasible set, interval hypergraph, hypertree, mixed hypergraph, unique colorability, gap in the chromatic spectrum, perfect hypergraph.
1571-0653/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2006.06.026
274
C. Bujtás, Z. Tuza / Electronic Notes in Discrete Mathematics 24 (2006) 273–275
Summary of talk We investigate vertex colorings of hypergraphs such that, for each edge, a lower and an upper bound are given on the number of colors occurring on the vertices of the edge. This generalizes the concept of ‘mixed hypergraph’ introduced by Voloshin [5], as ‘D-edge’ means lower bound 2, while ‘C-edge’ assumes that the lower bound is 1 and the upper bound is one smaller than the cardinality of the edge; cf. [6] as a comprehensive source of information. The ‘bi-edges’ occurring in the theory of mixed hypergraphs are represented in the frame of ‘mixed coloring’ in a natural, unified way. Our model also extends the one considered by Drgas-Burchardt and Lazuka [3] where just a lower bound not exceeding the edge size is given on the number of colors in each edge. In the talk we give a brief summary of results presented in two recent manuscripts, the first one [1] discussing general observations while the second one [2] concentrating on the major role of hypertrees (hypergraphs whose edges are vertex sets of subtrees of a tree). We mostly consider problems concerning the chromatic spectrum, which is the sequence whose k-th element is the number of allowed colorings with precisely k colors. In particular, the lower and upper chromatic number — the minimum and maximum number of colors in a coloring — are of great interest. We also prove complexity results much stronger than those valid for mixed hypergraphs, e.g. concerning unique colorability. A related but somewhat different generalization of mixed hypergraphs is introduced, too, by imposing conditions on the number of occurrences of each color in any edge. Similarly to the model described above, it is allowed that for different edges the coloring restrictions be different. Some facts remain valid for ‘pattern hypergraphs’ as well, a structure class introduced and studied recently by Dvoˇr´ak et al. in [4].
1
Research supported in part by the Hungarian Scientific Research Fund, OTKA grant T-049613. 2 Email: [email protected] 3 Email: [email protected]
C. Bujtás, Z. Tuza / Electronic Notes in Discrete Mathematics 24 (2006) 273–275
275
References [1] Bujt´as, Cs., and Zs. Tuza, Hypergraph mixed coloring, I: General results, preprint, 2006. [2] Bujt´as, Cs., and Zs. Tuza, Hypergraph mixed coloring, II: Interval hypergraphs and hypertrees, preprint, 2006. [3] Drgas-Burchardt, E., and E. Lazuka, On chromatic polynomials of hypergraphs, preprint, 2005. [4] Dvoˇr´ak, Z., J. K´ara, D. Kr´al’, and O. Pangr´ ac, Pattern hypergraphs, preprint, 2004. [5] Voloshin, V. I., On the upper chromatic number of a hypergraph, Australasian J. Combin. 11 (1995), 25–45. [6] Voloshin, V. I., “Coloring Mixed Hypergraphs – Theory, Algorithms and Applications,” Fields Institute Monographs 17, Amer. Math. Soc., Providence, R.I., 2002.