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 vertex Ramsey graphs and minimal forbidden subgraphs

AbstractLet P be a property of graphs. A graph G is vertex (P,k)-colourable if the vertex set V(G) of G can be partitioned into k sets V1,V2,…,Vk such that the subgraph G[Vi] of G belongs to P, i=1,2,…,k. If P is a hereditary property, then the set of minimal forbidden subgraphs of P is defined as follows: F(P)={G:G∉PbuteachpropersubgraphHofGbelongstoP}. In this paper we investigate the property On: each component of G has at most n+1 vertices. We construct minimal forbidden subgraphs for the property (Onk) “to be (On,k)-colourable”.We write G→v(H)k, k⩾2, if for each k-colouring V1,V2,…,Vk of a graph G there exists i, 1⩽i⩽k, such that the graph induced by the set Vi contains H as a subgraph. A graph G is called (H)k-vertex Ramsey minimal if G→v(H)k, but G′↛v(H)k for any proper subgraph G′ of G. The class of (P3)k-vertex Ramsey minimal graphs is investigated

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"

Unique factorization theorem

A property of graphs is any class of graphs closed under isomorphism. A property of graphs is induced-hereditary and additive if it is closed under taking induced subgraphs and disjoint unions of graphs, respectively. Let ₁,₂, ...,ₙ be properties of graphs. A graph G is (₁,₂,...,ₙ)-partitionable (G has property ₁ º₂ º... ºₙ) if the vertex set V(G) of G can be partitioned into n sets V₁,V₂,..., Vₙ such that the subgraph $G[V_i]$ of G induced by V_i belongs to $_i$; i = 1,2,...,n. A property is said to be reducible if there exist properties ₁ and ₂ such that = ₁ º₂; otherwise the property is irreducible. We prove that every additive and induced-hereditary property is uniquely factorizable into irreducible factors. Moreover the unique factorization implies the existence of uniquely (₁,₂, ...,ₙ)-partitionable graphs for any irreducible properties ₁,₂, ...,ₙ

Prime ideals in the lattice of additive induced-hereditary graph properties

An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove that the prime ideals of the lattice of all additive induced-hereditary properties are divided into two groups, determined either by a set of excluded join-irreducible properties or determined by a set of excluded properties with infinite join-decomposability number. We provide non-trivial examples of each type