A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set = E₁,...,Eₘ, together with integers and satisfying for each i = 1,...,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge satisfies . The hypergraph ℋ is colorable if it admits at least one proper coloring. We consider hypergraphs ℋ over a “host graph”, that means a graph G on the same vertex set X as ℋ, such that each induces a connected subgraph in G. In the current setting we fix a graph or multigraph G₀, and assume that the host graph G is obtained by some sequence of edge subdivisions, starting from G₀. The colorability problem is known to be NP-complete in general, and also when restricted to 3-uniform “mixed hypergraphs”, i.e., color-bounded hypergraphs in which and holds for all i ≤ m. We prove that for every fixed graph G₀ and natural number r, colorability is decidable in polynomial time over the class of r-uniform hypergraphs (and more generally of hypergraphs with for all 1 ≤ i ≤ m) having a host graph G obtained from G₀ by edge subdivisions. Stronger bounds are derived for hypergraphs for which G₀ is a tree.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1541,
author = {Csilla Bujt\'as and Zsolt Tuza and Vitaly Voloshin},
title = {Color-bounded hypergraphs, V: host graphs and subdivisions},
journal = {Discussiones Mathematicae Graph Theory},
volume = {31},
year = {2011},
pages = {223-238},
zbl = {1234.05078},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1541}
}
Csilla Bujtás; Zsolt Tuza; Vitaly Voloshin. Color-bounded hypergraphs, V: host graphs and subdivisions. Discussiones Mathematicae Graph Theory, Tome 31 (2011) pp. 223-238. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1541/
[000] [1] Cs. Bujtás and Zs. Tuza, Mixed colorings of hypergraphs, Electronic Notes in Discrete Math. 24 (2006) 273-275, doi: 10.1016/j.endm.2006.06.026.
[001] [2] Cs. Bujtás and Zs. Tuza, Uniform mixed hypergraphs: The possible numbers of colors, Graphs and Combinatorics 24 (2008) 1-12, doi: 10.1007/s00373-007-0765-5.
[002] [3] Cs. Bujtás and Zs. Tuza, Color-bounded hypergraphs, I: General results, Discrete Math. 309 (2009) 4890-4902, doi: 10.1016/j.disc.2008.04.019. | Zbl 1210.05088
[003] [4] Cs. Bujtás and Zs. Tuza, Color-bounded hypergraphs, II: Interval hypergraphs and hypertrees, Discrete Math. 309 (2009) 6391-6401, doi: 10.1016/j.disc.2008.10.023. | Zbl 1225.05189
[004] [5] Cs. Bujtás and Zs. Tuza, Color-bounded hypergraphs, III: Model comparison, Appl. Anal. and Discrete Math. 1 (2007) 36-55.
[005] [6] Cs. Bujtás and Zs. Tuza, Color-bounded hypergraphs, IV: Stable colorings of hypertrees, Discrete Math. 310 (2010) 1463-1474, doi: 10.1016/j.disc.2009.07.014. | Zbl 1219.05055
[006] [7] Cs. Bujtás and Zs. Tuza, Coloring intervals with four types of constraints, in: 6th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications, A. Frank et al., Eds. (Budapest, Hungary, May 16-19, 2009) 393-401.
[007] [8] E. Drgas-Burchardt and E. Łazuka, On chromatic polynomials of hypergraphs, Appl. Math. Letters 20 (2007) 1250-1254, doi: 10.1016/j.aml.2007.02.006. | Zbl 1137.05030
[008] [9] T. Jiang, D. Mubayi, Zs. Tuza, V.I. Voloshin and D. West, The chromatic spectrum of mixed hypergraphs, Graphs and Combinatorics 18 (2002) 309-318, doi: 10.1007/s003730200023. | Zbl 0994.05063
[009] [10] D. Král', J. Kratochví l, A. Proskurowski and H.-J. Voss, Coloring mixed hypertrees, in: 26th Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science 1928 (Springer-Verlag, 2000) 279-289.
[010] [11] D. Král, J. Kratochvíl, and H.-J. Voss, Mixed hypercacti, Discrete Math. 286 (2004) 99-113, doi: 10.1016/j.disc.2003.11.051. | Zbl 1064.05060
[011] [12] Zs. Tuza and V. Voloshin, Uncolorable mixed hypergraphs, Discrete Appl. Math. 99 (2000) 209-227, doi: 10.1016/S0166-218X(99)00134-1. | Zbl 0943.05035
[012] [13] Zs. Tuza and V. Voloshin, Problems and results on colorings of mixed hypergraphs, Horizon of Combinatorics (E. Gyori et al., Eds.), Bolyai Society Mathematical Studies 17 (Springer-Verlag, 2008) 235-255. | Zbl 1201.05040
[013] [14] V. Voloshin, The mixed hypergraphs, Computer Science Journal of Moldova 1 (1993) 45-52.
[014] [15] V. Voloshin, On the upper chromatic number of a hypergraph, Australasian J. Combin. 11 (1995) 25-45. | Zbl 0827.05027
[015] [16] V.I. Voloshin, Coloring Mixed Hypergraphs: Theory, Algorithms and Applications, Fields Institute Monographs 17 Amer. Math. Soc., 2002.