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

Csilla Bujtás; Zsolt Tuza; Vitaly Voloshin

Discussiones Mathematicae Graph Theory (2011)

- Volume: 31, Issue: 2, page 223-238
- ISSN: 2083-5892

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topCsilla Bujtás, Zsolt Tuza, and Vitaly Voloshin. "Color-bounded hypergraphs, V: host graphs and subdivisions." Discussiones Mathematicae Graph Theory 31.2 (2011): 223-238. <http://eudml.org/doc/270851>.

@article{CsillaBujtás2011,

abstract = {A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set = 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.},

author = {Csilla Bujtás, Zsolt Tuza, Vitaly Voloshin},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {mixed hypergraph; color-bounded hypergraph; vertex coloring; arboreal hypergraph; hypertree; feasible set; host graph; edge subdivision},

language = {eng},

number = {2},

pages = {223-238},

title = {Color-bounded hypergraphs, V: host graphs and subdivisions},

url = {http://eudml.org/doc/270851},

volume = {31},

year = {2011},

}

TY - JOUR

AU - Csilla Bujtás

AU - Zsolt Tuza

AU - Vitaly Voloshin

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

JO - Discussiones Mathematicae Graph Theory

PY - 2011

VL - 31

IS - 2

SP - 223

EP - 238

AB - A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set = 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.

LA - eng

KW - mixed hypergraph; color-bounded hypergraph; vertex coloring; arboreal hypergraph; hypertree; feasible set; host graph; edge subdivision

UR - http://eudml.org/doc/270851

ER -

## References

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