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Unique factorization theorem

Peter Mihók — 2000

Discussiones Mathematicae Graph Theory

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 Vi belongs to i ; i = 1,2,...,n. A property is said to be reducible...

Fractional Q-Edge-Coloring of Graphs

Július Czap;  Peter Mihók — 2013

Discussiones Mathematicae Graph Theory

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let [...] be an additive hereditary property of graphs. A [...] -edge-coloring of a simple graph is an edge coloring in which the edges colored with the same color induce a subgraph of property [...] . In this paper we present some results on fractional [...] -edge-colorings. We determine the fractional [...] -edge chromatic number for matroidal properties of graphs.

Gallai's innequality for critical graphs of reducible hereditary properties

Peter Mihók;  Riste Skrekovski — 2001

Discussiones Mathematicae Graph Theory

In this paper Gallai’s inequality on the number of edges in critical graphs is generalized for reducible additive induced-hereditary properties of graphs in the following way. Let ₁ , ₂ , . . . , ₖ (k ≥ 2) be additive induced-hereditary properties, = ₁ ∘ ₂ ∘ . . . ∘ ₖ and δ = ∑ i = 1 k δ ( i ) . Suppose that G is an -critical graph with n vertices and m edges. Then 2m ≥ δn + (δ-2)/(δ²+2δ-2)*n + (2δ)/(δ²+2δ-2) unless = ² or G = K δ + 1 . The generalization of Gallai’s inequality for -choice critical graphs is also presented.

On uniquely partitionable relational structures and object systems

Jozef Bucko;  Peter Mihók — 2006

Discussiones Mathematicae Graph Theory

We introduce object systems as a common generalization of graphs, hypergraphs, digraphs and relational structures. Let C be a concrete category, a simple object system over C is an ordered pair S = (V,E), where E = A₁,A₂,...,Aₘ is a finite set of the objects of C, such that the ground-set V ( A i ) of each object A i ∈ E is a finite set with at least two elements and V ⊇ ⋃ i = 1 m V ( A i ) . To generalize the results on graph colourings to simple object systems we define, analogously as for graphs, that an additive induced-hereditary...

On infinite uniquely partitionable graphs and graph properties of finite character

Jozef Bucko;  Peter Mihók — 2009

Discussiones Mathematicae Graph Theory

A graph property is any nonempty isomorphism-closed class of simple (finite or infinite) graphs. A graph property is of finite character if a graph G has a property if and only if every finite induced subgraph of G has a property . Let ₁,₂,...,ₙ be graph properties of finite character, a graph G is said to be (uniquely) (₁, ₂, ...,ₙ)-partitionable if there is an (exactly one) partition V₁, V₂, ..., Vₙ of V(G) such that G [ V i ] ∈ i for i = 1,2,...,n. Let us denote by ℜ = ₁ ∘ ₂ ∘ ... ∘ ₙ the class of all (₁,₂,...,ₙ)-partitionable...

Unique factorization theorem for object-systems

Peter Mihók;  Gabriel Semanišin — 2011

Discussiones Mathematicae Graph Theory

The concept of an object-system is a common generalization of simple graph, digraph and hypergraph. In the theory of generalised colourings of graphs, the Unique Factorization Theorem (UFT) for additive induced-hereditary properties of graphs provides an analogy of the well-known Fundamental Theorem of Arithmetics. The purpose of this paper is to present UFT for object-systems. This result generalises known UFT for additive induced-hereditary and hereditary properties of graphs and digraphs. Formal...

Fractional (P,Q)-Total List Colorings of Graphs

Arnfried Kemnitz;  Peter Mihók;  Margit Voigt — 2013

Discussiones Mathematicae Graph Theory

Let r, s ∈ N, r ≥ s, and P and Q be two additive and hereditary graph properties. A (P,Q)-total (r, s)-coloring of a graph G = (V,E) is a coloring of the vertices and edges of G by s-element subsets of Zr such that for each color i, 0 ≤ i ≤ r − 1, the vertices colored by subsets containing i induce a subgraph of G with property P, the edges colored by subsets containing i induce a subgraph of G with property Q, and color sets of incident vertices and edges are disjoint. The fractional (P,Q)-total...

Universality in Graph Properties with Degree Restrictions

Izak Broere;  Johannes Heidema;  Peter Mihók — 2013

Discussiones Mathematicae Graph Theory

Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in the m’th position of its binary expansion. It is well known that R is a universal graph in the set [...] of all countable graphs (since every graph in [...] is isomorphic to an induced subgraph of R). A brief overview of known universality results for some induced-hereditary subsets of [...] is provided. We then construct...

Graphs maximal with respect to hom-properties

Jan Kratochvíl;  Peter Mihók;  Gabriel Semanišin — 1997

Discussiones Mathematicae Graph Theory

For a simple graph H, →H denotes the class of all graphs that admit homomorphisms to H (such classes of graphs are called hom-properties). We investigate hom-properties from the point of view of the lattice of hereditary properties. In particular, we are interested in characterization of maximal graphs belonging to →H. We also provide a description of graphs maximal with respect to reducible hom-properties and determine the maximum number of edges of graphs belonging to →H.

On generalized list colourings of graphs

Mieczysław Borowiecki;  Izak Broere;  Peter Mihók — 1997

Discussiones Mathematicae Graph Theory

Vizing [15] and Erdős et al. [8] independently introduce the idea of considering list-colouring and k-choosability. In the both papers the choosability version of Brooks' theorem [4] was proved but the choosability version of Gallai's theorem [9] was proved independently by Thomassen [14] and by Kostochka et al. [11]. In [3] some extensions of these two basic theorems to (𝓟,k)-choosability have been proved. In this paper we prove some extensions of the well-known bounds for...

The order of uniquely partitionable graphs

Izak Broere;  Marietjie Frick;  Peter Mihók — 1997

Discussiones Mathematicae Graph Theory

Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition V₁,...,Vₙ of V(G) such that, for each i = 1,...,n, the subgraph of G induced by V i has property i . If a graph G has a unique (₁,...,ₙ)-partition we say it is uniquely (₁,...,ₙ)-partitionable. We establish best lower bounds for the order of uniquely (₁,...,ₙ)-partitionable graphs, for various choices of ₁,...,ₙ.

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

Amelie J. Berger;  Peter Mihók — 2003

Discussiones Mathematicae Graph Theory

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,...

Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties

Izak Broere;  Jozef Bucko;  Peter Mihók — 2002

Discussiones Mathematicae Graph Theory

Let ₁,₂,...,ₙ be graph properties, a graph G is said to be uniquely (₁,₂, ...,ₙ)-partitionable if there is exactly one (unordered) partition V₁,V₂,...,Vₙ of V(G) such that G [ V i ] ∈ i for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (₁,₂,...,ₙ)-partitionable graphs exist if and only if i and j are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ 1,2,...,n.

Generalized circular colouring of graphs

Peter Mihók;  Janka Oravcová;  Roman Soták — 2011

Discussiones Mathematicae Graph Theory

Let P be a graph property and r,s ∈ N, r ≥ s. A strong circular (P,r,s)-colouring of a graph G is an assignment f:V(G) → {0,1,...,r-1}, such that the edges uv ∈ E(G) satisfying |f(u)-f(v)| < s or |f(u)-f(v)| > r - s, induce a subgraph of G with the propery P. In this paper we present some basic results on strong circular (P,r,s)-colourings. We introduce the strong circular P-chromatic number of a graph and we determine the strong circular P-chromatic number of complete graphs for additive...

On universal graphs for hom-properties

Peter Mihók;  Jozef Miškuf;  Gabriel Semanišin — 2009

Discussiones Mathematicae Graph Theory

A graph property is any isomorphism closed class of simple graphs. For a simple finite graph H, let → H denote the class of all simple countable graphs that admit homomorphisms to H, such classes of graphs are called hom-properties. Given a graph property 𝓟, a graph G ∈ 𝓟 is universal in 𝓟 if each member of 𝓟 is isomorphic to an induced subgraph of G. In particular, we consider universal graphs in → H and we give a new proof of the existence of a universal graph in → H, for any finite graph...

Generalized Fractional and Circular Total Colorings of Graphs

Arnfried Kemnitz;  Massimiliano Marangio;  Peter Mihók;  Janka Oravcová;  Roman Soták — 2015

Discussiones Mathematicae Graph Theory

Let P and Q be additive and hereditary graph properties, r, s ∈ N, r ≥ s, and [ℤr]s be the set of all s-element subsets of ℤr. An (r, s)-fractional (P,Q)-total coloring of G is an assignment h : V (G) ∪ E(G) → [ℤr]s such that for each i ∈ ℤr the following holds: the vertices of G whose color sets contain color i induce a subgraph of G with property P, edges with color sets containing color i induce a subgraph of G with property Q, and the color sets of incident vertices and edges are disjoint. If...

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