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Uniquely partitionable graphs

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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 $G[V_i]$ induced by $V_i$ has property $𝓟_i$; 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.
Opis fizyczny
  • Department of Geometry and Algebra, P.J. Šafárik University, Jesenná 5, 041 54 Košice, Slovak Republic
  • Department of Mathematics, Applied Mathematics and Astronomy, University of South Africa, P.O. Box 392, Pretoria, 0001 South Africa
  • Department of Geometry and Algebra, P.J. Šafárik University, Jesenná 5, 041 54 Košice, Slovak Republic
  • Department of Geometry and Algebra, P.J. Šafárik University, Jesenná 5, 041 54 Košice, Slovak Republic
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