Discussiones
Mathematicae Graph Theory 22(1) (2002) 31-37
DOI: https://doi.org/10.7151/dmgt.1156
CRITERIA FOR OF THE EXISTENCE OF UNIQUELY PARTITIONABLE GRAPHS WITH RESPECT TO ADDITIVE INDUCED-HEREDITARY PROPERTIES
Izak Broere
Department of Mathematics |
Jozef Bucko
Department of Applied Mathematics |
Peter Mihók
Department of Applied Mathematics |
Abstract
Let P1,P2,...,Pn be graph properties, a graph G is said to be uniquely (P1, P2, …,Pn)-partitionable if there is exactly one (unordered) partition {V1,V2, …,Vn} of V(G) such that G[Vi] ∈ Pi for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (P1, P2, …,Pn)-partitionable graphs exist if and only if Pi and Pj are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ {1,2,...,n}.Keywords: induced-hereditary properties, reducibility, divisibility, uniquely partitionable graphs.
2000 Mathematics Subject Classification: 05C15, 05C75.
Received 8 August 2000References
[1]
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.
[3]
I. Broere and J. Bucko, Divisibility in additive hereditary graph properties and
uniquely partitionable graphs, Tatra Mt. Math. Publ. 18 (1999) 79-87.
[4]
J. Bucko, M. Frick, P. Mihók and R. Vasky, Uniquely partitionable
graphs, Discuss. Math. Graph Theory 17 (1997) 103-114, doi: 10.7151/dmgt.1043.
[5]
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.
[6]
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.
[7]
P. Mihók, Unique factorization theorem, Discuss. Math. Graph Theory 20
(2000) 143-154, doi: 10.7151/dmgt.1114.
Revised 2 July 2001
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