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Tytuł artykułu

Color-bounded hypergraphs, V: host graphs and subdivisions

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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.
Opis fizyczny
  • Department of Computer Science and Systems Technology, University of Pannonia, H-8200 Veszprém, Egyetem u. 10, Hungary
  • 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
  • Department of Mathematics, Physics,, Computer Science and Geomatics, Troy University, Troy, AL 36082, USA
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  • [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.
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  • [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.
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