# Unique factorization theorem

• Volume: 20, Issue: 1, page 143-154
• ISSN: 2083-5892

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## Abstract

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A property of graphs is any class of graphs closed under isomorphism. A property of graphs is induced-hereditary and additive if it is closed under taking induced subgraphs and disjoint unions of graphs, respectively. Let ₁,₂, ...,ₙ be properties of graphs. A graph G is (₁,₂,...,ₙ)-partitionable (G has property ₁ º₂ º... ºₙ) if the vertex set V(G) of G can be partitioned into n sets V₁,V₂,..., Vₙ such that the subgraph $G\left[{V}_{i}\right]$ of G induced by Vi belongs to ${}_{i}$; i = 1,2,...,n. A property is said to be reducible if there exist properties ₁ and ₂ such that = ₁ º₂; otherwise the property is irreducible. We prove that every additive and induced-hereditary property is uniquely factorizable into irreducible factors. Moreover the unique factorization implies the existence of uniquely (₁,₂, ...,ₙ)-partitionable graphs for any irreducible properties ₁,₂, ...,ₙ.

## How to cite

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Peter Mihók. "Unique factorization theorem." Discussiones Mathematicae Graph Theory 20.1 (2000): 143-154. <http://eudml.org/doc/270654>.

@article{PeterMihók2000,
abstract = {A property of graphs is any class of graphs closed under isomorphism. A property of graphs is induced-hereditary and additive if it is closed under taking induced subgraphs and disjoint unions of graphs, respectively. Let ₁,₂, ...,ₙ be properties of graphs. A graph G is (₁,₂,...,ₙ)-partitionable (G has property ₁ º₂ º... ºₙ) if the vertex set V(G) of G can be partitioned into n sets V₁,V₂,..., Vₙ such that the subgraph $G[V_i]$ of G induced by Vi belongs to $_i$; i = 1,2,...,n. A property is said to be reducible if there exist properties ₁ and ₂ such that = ₁ º₂; otherwise the property is irreducible. We prove that every additive and induced-hereditary property is uniquely factorizable into irreducible factors. Moreover the unique factorization implies the existence of uniquely (₁,₂, ...,ₙ)-partitionable graphs for any irreducible properties ₁,₂, ...,ₙ.},
author = {Peter Mihók},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {induced-hereditary; additive property of graphs; reducible property of graphs; unique factorization; uniquely partitionable graphs; generating sets; additive property; reducible property; partitionable graphs; partition; induced-hereditary property; factors},
language = {eng},
number = {1},
pages = {143-154},
title = {Unique factorization theorem},
url = {http://eudml.org/doc/270654},
volume = {20},
year = {2000},
}

TY - JOUR
AU - Peter Mihók
TI - Unique factorization theorem
JO - Discussiones Mathematicae Graph Theory
PY - 2000
VL - 20
IS - 1
SP - 143
EP - 154
AB - A property of graphs is any class of graphs closed under isomorphism. A property of graphs is induced-hereditary and additive if it is closed under taking induced subgraphs and disjoint unions of graphs, respectively. Let ₁,₂, ...,ₙ be properties of graphs. A graph G is (₁,₂,...,ₙ)-partitionable (G has property ₁ º₂ º... ºₙ) if the vertex set V(G) of G can be partitioned into n sets V₁,V₂,..., Vₙ such that the subgraph $G[V_i]$ of G induced by Vi belongs to $_i$; i = 1,2,...,n. A property is said to be reducible if there exist properties ₁ and ₂ such that = ₁ º₂; otherwise the property is irreducible. We prove that every additive and induced-hereditary property is uniquely factorizable into irreducible factors. Moreover the unique factorization implies the existence of uniquely (₁,₂, ...,ₙ)-partitionable graphs for any irreducible properties ₁,₂, ...,ₙ.
LA - eng
KW - induced-hereditary; additive property of graphs; reducible property of graphs; unique factorization; uniquely partitionable graphs; generating sets; additive property; reducible property; partitionable graphs; partition; induced-hereditary property; factors
UR - http://eudml.org/doc/270654
ER -

## References

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1. [1] D. Achlioptas, J.I. Brown, D.G. Corneil and M.S.O. Molloy, The existence of uniquely -G colourable graphs, Discrete Math. 179 (1998) 1-11, doi: 10.1016/S0012-365X(97)00022-8. Zbl0885.05062
2. [2] A. Berger, Reducible properties have infinitely many minimal forbidden subgraphs, manuscript.
3. [3] B. Bollobás and A.G. Thomason, Hereditary and monotone properties of graphs, in: R.L. Graham and J. Nesetril, eds., The mathematics of Paul Erdős, II, Algorithms and Combinatorics vol. 14 (Springer-Verlag, 1997), 70-78. Zbl0866.05030
4. [4] M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, Survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997) 5-50, doi: 10.7151/dmgt.1037. Zbl0902.05026
5. [5] I. Broere, J. Bucko, Divisibility in additive hereditary graph properties and uniquely partitionable graphs, Tatra Mountains Math. Publications 18 (1999) 79-87. Zbl0951.05034
6. [6] E.J. Cockayne, Color clasess for r-graphs, Canad. Math. Bull. 15 (3) (1972) 349-354, doi: 10.4153/CMB-1972-063-2. Zbl0254.05106
7. [7] R.L. Graham, M. Grötschel and L. Lovász, Handbook of combinatorics (Elsevier Science B.V., Amsterdam, 1995). Zbl0833.05001
8. [8] T.R. Jensen and B. Toft, Graph colouring problems (Wiley-Interscience Publications, New York, 1995). Zbl0971.05046
9. [9] J. Kratochvíl, P. Mihók, Hom-properties are uniquely factorizable into irreducible factors, Discrete Math. 213 (2000) 189-194, doi: 10.1016/S0012-365X(99)00179-X. Zbl0949.05025
10. [10] P. Mihók, Additive hereditary properties and uniquely partitionable graphs, in: M. Borowiecki and Z. Skupień, eds., Graphs, hypergraphs and matroids (Zielona Góra, 1985), 49-58. Zbl0623.05043
11. [11] P. Mihók and R. Vasky, On the factorization of reducible properties of graphs into irreducible factors, Discuss. Math. Graph Theory 15 (1995) 195-203, doi: 10.7151/dmgt.1017. Zbl0845.05076
12. [12] P. Mihók, Reducible properties and uniquely partitionable graphs, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 49 (1999) 213-218. Zbl0937.05042
13. [13] P. Mihók, G. Semanišin and R. Vasky, Additive and Hereditary Properties of Graphs are Uniquely Factorizable into Irreducible Factors, J. Graph Theory 33 (2000) 44-53, doi: 10.1002/(SICI)1097-0118(200001)33:1<44::AID-JGT5>3.0.CO;2-O Zbl0942.05056
14. [14] G. Semanišin, On generating sets of induced-hereditary properties, manuscript. Zbl1018.05089

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