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## Discussiones Mathematicae Graph Theory

2011 | 31 | 2 | 223-238
Tytuł artykułu

### Color-bounded hypergraphs, V: host graphs and subdivisions

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set 𝓔 = {E₁,...,Eₘ}, together with integers $s_i$ and $t_i$ satisfying $1 ≤ s_i ≤ t_i ≤ |E_i|$ for each i = 1,...,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge $E_i$ satisfies $s_i ≤ |φ(E_i)| ≤ t_i$. 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 $E_i$ 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 $|E_i| = 3$ and $1 ≤ s_i ≤ 2 ≤ t_i ≤ 3$ 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 $|E_i| ≤ r$ 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.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
223-238
Opis fizyczny
Daty
wydano
2011
otrzymano
2009-11-23
poprawiono
2010-07-14
zaakceptowano
2010-07-14
Twórcy
autor
• Department of Computer Science and Systems Technology, University of Pannonia, H-8200 Veszprém, Egyetem u. 10, Hungary
autor
• Department of Computer Science and Systems Technology, University of Pannonia, H-8200 Veszprém, Egyetem u. 10, Hungary
• Computer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest, Kende u. 13-17, Hungary
autor
• Department of Mathematics, Physics,, Computer Science and Geomatics, Troy University, Troy, AL 36082, USA
Bibliografia
• [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.
• [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.
• [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.
• [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.
• [5] Cs. Bujtás and Zs. Tuza, Color-bounded hypergraphs, III: Model comparison, Appl. Anal. and Discrete Math. 1 (2007) 36-55.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [12] Zs. Tuza and V. Voloshin, Uncolorable mixed hypergraphs, Discrete Appl. Math. 99 (2000) 209-227, doi: 10.1016/S0166-218X(99)00134-1.
• [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.
• [14] V. Voloshin, The mixed hypergraphs, Computer Science Journal of Moldova 1 (1993) 45-52.
• [15] V. Voloshin, On the upper chromatic number of a hypergraph, Australasian J. Combin. 11 (1995) 25-45.
• [16] V.I. Voloshin, Coloring Mixed Hypergraphs: Theory, Algorithms and Applications, Fields Institute Monographs 17 Amer. Math. Soc., 2002.
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Bibliografia
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