19 research outputs found
Remarks on the existence of uniquely partitionable planar graphs
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
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
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
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
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