DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory  17(1) (1997)   103-113
DOI: https://doi.org/10.7151/dmgt.1043

UNIQUELY PARTITIONABLE GRAPHS

Jozef Bucko

Department of Geometry and Algebra, P.J. Šafárik University
Jesenná 5, 041 54 Košice, Slovak Republic
e-mail: bucko@duro.upjs.sk

Marietjie Frick

Department of Mathematics, Applied Mathematics and Astronomy
University of South Africa, P.O. Box 392, Pretoria, 0001 South Africa
e-mail: frickm@risc5.unisa.ac.za

Peter Mihók and Roman Vasky

Department of Geometry and Algebra, P.J. Šafárik University
Jesenná 5, 041 54 Košice, Slovak Republic
e-mail: mihok@Košice.upjs.sk
e-mail: vasky@duro.upjs.sk

Abstract

Let P1,…,Pn be properties of graphs. A (P1,…,Pn)-partition of a graph G is a partition of the vertex set V(G) into subsets V1, …,Vn such that the subgraph G[Vi] induced by Vi has property Pi; i = 1,…,n. A graph G is said to be uniquely (P1, …,Pn)-partitionable if G has exactly one (P1,…,Pn)-partition. A property P is called hereditary if every subgraph of every graph with property P also has property P. If every graph that is a disjoint union of two graphs that have property P also has property P, then we say that P is additive. A property P is called degenerate if there exists a bipartite graph that does not have property P. In this paper, we prove that if P1,…, Pn are degenerate, additive, hereditary properties of graphs, then there exists a uniquely (P1,…,Pn)-partitionable graph.

Keywords: hereditary property of graphs, additivity, reducibility, vertex partition.

1991 Mathematics Subject Classification: 05C15, 05C70.

References

[1] J.A. Andrews and M.S. Jacobson, On a generalization of chromatic number, Congressus Numerantium 47 (1985) 33-48.
[2] G. Benadé, I. Broere and J.I. Brown, A construction of uniquely C4-free colourable graphs, Questiones Mathematicae 13 (1990) 259-264, doi: 10.1080/16073606.1990.9631616.
[3] G. Benadé, I. Broere, B. Jonck and M. Frick, Uniquely (m,k)τ-colourable graphs and k-τ-saturated graphs, Discrete Math. 162 (1996) 13-22, doi: 10.1016/0012-365X(95)00301-C.
[4] B. Bollobás and N. Sauer, Uniquely colourable graphs with large girth, Canad. J. Math. 28 (1976) 1340-1344, doi: 10.4153/CJM-1976-133-5.
[5] B. Bollobás and A.G. Thomason, Uniquely partitionable graphs, J. London Math. Soc. 16 (1977) 403-410, doi: 10.1112/jlms/s2-16.3.403.
[6] M. Borowiecki and P. Mihók, Hereditary properties of graphs, in: V.R. Kulli, eds., Advances in Graph Theory, (Vishwa International Publication, Gulbarga 1991) 42-69.
[7] I. Broere and M. Frick, On the order of uniquely (k,m)-colourable graphs, Discrete Math. 82 (1990) 225-232, doi: 10.1016/0012-365X(90)90200-2.
[8] J.I. Brown and D.G. Corneil, On generalized graph colourings, J. Graph Theory 11 (1987) 86-99, doi: 10.1002/jgt.3190110113.
[9] J.I. Brown, D. Kelly, J. Schoenheim and R.E. Woodrow, Graph coloring satisfying restraints, Discrete Math. (1990) 123-143, doi: 10.1016/0012-365X(90)90113-V.
[10] S.A. Burr and M.S. Jacobson, On inequalities involving vertex-partition parameters of graphs, Congressus Numerantium 70 (1990) 159-170.
[11] L.J. Cowen, R.H. Cowen and D.R. Woodall, Defective colorings of graphs in surfaces; partitions into subgraphs of bounded valency, J. Graph Theory 10 (1986) 187-195, doi: 10.1002/jgt.3190100207.
[12] M. Frick, A survey of (m,k)-colourings, in: J. Gimbel c.a, eds., Quo Vadis,Graph Theory? Annals of Discrete Mathematics 55 (North-Holland, Amsterdam, 1993) 45-58.
[13] M. Frick and M.A. Henning, Extremal results on defective colorings of graph, Discrete Math. 126 (1994) 151-158, doi: 10.1016/0012-365X(94)90260-7.
[14] D. Gernet, Forbidden and unavoidable subgraphs, Ars Combinatoria 27 (1989) 165-176.
[15] R.L. Graham, M. Grötschel and L. Lovász, Handbook of Combinatorics (Elsevier Science B.V., Amsterdam 1995).
[16] D.L. Greenwell, R.L. Hemminger and J. Klerlein, Forbidden subgraphs, in: Proc. 4th S-E Conf. Combinatorics, Graph Theory and Computing, (Utilitas Math., Winnipeg, Man., 1973) 389-394.
[17] F. Harary, S.T. Hedetniemi and R.W. Robinson, Uniquely colourable graphs, J. Combin. Theory 6 (1969) 264-270, doi: 10.1016/S0021-9800(69)80086-4.
[18] G. Chartrand, D. Geller and S. Hedetniemi, Graphs with forbidden subgraphs, J. Combin. Theory (B) 10 (1971) 12-41, doi: 10.1016/0095-8956(71)90065-7.
[19] G. Chartrand and J.P. Geller, Uniquely colourable planar graphs, J. Combin. Theory 6 (1969) 271-278, doi: 10.1016/S0021-9800(69)80087-6.
[20] T.R. Jensen and B. Toft, Graph Colouring Problems (Wiley-Interscience Publications, New York 1995).
[21] R.P. Jones, Hereditary properties and P-chromatic number, in: T.P. McDonough and V.C. Marron, eds., Combinatorics, Proc. Brit. Comb. Conf. (London Math. Soc. Lecture Note Ser., No.13, Cambridge Univ. Press, London 1974) 83-88.
[22] L. Lovász, On decomposition of graphs, Studia Sci. Math. Hungar 1 (1966) 237-238.
[23] 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.
[24] J. Mitchem, Maximal k-degenerate graphs, Utilitas Math. 11 (1977) 101-106.
[25] F.S. Roberts, From garbage to rainbows: generalizations of graph colouring and their applications, in: Y. Alavi, G. Chartrand, O.R. Oellermann, and A.J. Schwenk, eds., Graph Theory, Combinatorics and Applications, (Wiley, New York, 1991) 1031-1052.
[26] F.S. Roberts, New directions in graph theory (with an emphasis on the role of applications), in: J. Gimbel, J.W. Kennedy, and L.V. Quintas, eds., Quo Vadis Graph Theory, (North-Holland, Amsterdam, 1993) 13-44.
[27] J.M.S. Simoes-Pereira, Joins of n-degenerate graphs and uniquely (m,n)-partitionable graphs, J. Combin. Theory (B) 21 (1976) 21-29, doi: 10.1016/0095-8956(76)90023-X.
[28] J.M.S. Simoes-Pereira, On graphs uniquely partitionable into n-degenerate subgraphs, in: Infinite and Finite Sets Colloquia Math. Soc. J. Bólyai 10 (North-Holland, Amsterdam, 1975) 1351-1364.
[29] M. Simonovits, Extremal graph theory, in: L.W. Beineke and R.J. Wilson, eds., Selected Topics in Graph Theory, 2 (Academic Press, London, 1983) 161-200.
[30] M. Simonovits, Extremal graph problems and graph products, in: Studies in Pure Math. (dedicated to the memory of P. Turán) (1983) 669-680.
[31] M. Weaver and D.B. West, Relaxed chromatic numbers of graphs, Graphs and Combinatorics 10 (1994) 75-93, doi: 10.1007/BF01202473.
[32] D. Woodall, Improper colorings of graphs, in: R. Nelson and R.J. Wilson, eds., Graph Colorings (Longman, New York, 1990) 45-64.
[33] X. Zhu, Uniquely H-colorable graphs with large girth, J. Graph Theory 23 (1996) 33-41, doi: 10.1002/(SICI)1097-0118(199609)23:1<33::AID-JGT3>3.0.CO;2-L.


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