19 research outputs found

    Remarks on the existence of uniquely partitionable planar graphs

    Get PDF
    We consider the problem of the existence of uniquely partitionable planar graphs. We survey some recent results and we prove the nonexistence of uniquely (D1,D1)-partitionable planar graphs with respect to the property D1 "to be a forest"

    Minimal reducible bounds for the class of k-degenerate graphs

    Get PDF
    AbstractLet (La,⊆) be the lattice of hereditary and additive properties of graphs. A reducible property R∈La is called minimal reducible bound for a property P∈La if in the interval (P,R) of the lattice La, there are only irreducible properties. We prove that the set B(Dk)={Dp∘Dq:k=p+q+1} is the covering set of minimal reducible bounds for the class Dk of all k-degenerate graphs

    Minimal reducible bounds for induced-hereditary properties

    Get PDF
    AbstractLet (Ma,⊆) and (La,⊆) be the lattices of additive induced-hereditary properties of graphs and additive hereditary properties of graphs, respectively. A property R∈Ma (∈La) is called a minimal reducible bound for a property P∈Ma (∈La) if in the interval (P,R) of the lattice Ma (La) there are only irreducible properties. The set of all minimal reducible bounds of a property P∈Ma in the lattice Ma we denote by BM(P). Analogously, the set of all minimal reducible bounds of a property P∈La in La is denoted by BL(P).We establish a method to determine minimal reducible bounds for additive degenerate induced-hereditary (hereditary) properties of graphs. We show that this method can be successfully used to determine already known minimal reducible bounds for k-degenerate graphs and outerplanar graphs in the lattice La. Moreover, in terms of this method we describe the sets of minimal reducible bounds for partial k-trees and the graphs with restricted order of components in La and k-degenerate graphs in Ma

    The property of k-colourable graphs is uniquely decomposable

    Get PDF
    An additive hereditary graph property is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If P1; : : : ;Pn are graph properties, then a (P1; : : : ;Pn)-decomposition of a graph G is a partition E1; : : : ;En of E(G) such that G[Ei], the subgraph of G induced by Ei, is in Pi, for i = 1; : : : ; n. The sum of the properties P1; : : : ;Pn is the property P1 Pn = fG 2 I : G has a (P1; : : : ;Pn)-decompositiong. A property P is said to be decomposable if there exist non-trivial additive hereditary properties P1 and P2 such that P = P1 P2. A property is uniquely decomposable if, apart from the order of the factors, it can be written as a sum of indecomposable properties in only one way. We show that not all properties are uniquely decomposable; however, the property of k-colourable graphs Ok is a uniquely decomposable property. Keywords: graph property, decomposable propertyhttp://www.elsevier.com/locate/dischb201

    Unsolved Problems in Group Theory. The Kourovka Notebook

    Full text link
    This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 19th edition, which contains 111 new problems and a number of comments on about 1000 problems from the previous editions.Comment: A few new solutions and references have been added or update
    corecore