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Acyclic 6-Colouring of Graphs with Maximum Degree 5 and Small Maximum Average Degree

Anna Fiedorowicz — 2013

Discussiones Mathematicae Graph Theory

A k-colouring of a graph G is a mapping c from the set of vertices of G to the set {1, . . . , k} of colours such that adjacent vertices receive distinct colours. Such a k-colouring is called acyclic, if for every two distinct colours i and j, the subgraph induced by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, every cycle in G has at least three distinct colours. Acyclic colourings were introduced by Gr¨unbaum in 1973, and since then have...

On partitions of hereditary properties of graphs

Mieczysław Borowiecki;  Anna Fiedorowicz — 2006

Discussiones Mathematicae Graph Theory

In this paper a concept 𝓠-Ramsey Class of graphs is introduced, where 𝓠 is a class of bipartite graphs. It is a generalization of well-known concept of Ramsey Class of graphs. Some 𝓠-Ramsey Classes of graphs are presented (Theorem 1 and 2). We proved that 𝓣₂, the class of all outerplanar graphs, is not 𝓓₁-Ramsey Class (Theorem 3). This results leads us to the concept of acyclic reducible bounds for a hereditary property 𝓟 . For 𝓣₂ we found two bounds (Theorem 4). An improvement, in some sense,...

Acyclic reducible bounds for outerplanar graphs

Mieczysław Borowiecki;  Anna Fiedorowicz;  Mariusz Hałuszczak — 2009

Discussiones Mathematicae Graph Theory

For a given graph G and a sequence ₁, ₂,..., ₙ of additive hereditary classes of graphs we define an acyclic (₁, ₂,...,Pₙ)-colouring of G as a partition (V₁, V₂,...,Vₙ) of the set V(G) of vertices which satisfies the following two conditions: 1. G [ V i ] ∈ i for i = 1,...,n, 2. for every pair i,j of distinct colours the subgraph induced in G by the set of edges uv such that u ∈ V i and v ∈ V j is acyclic. A class R = ₁ ⊙ ₂ ⊙ ... ⊙ ₙ is defined as the set of the graphs having an acyclic (₁, ₂,...,Pₙ)-colouring. If ⊆ R,...

Maximal hypergraphs with respect to the bounded cost hereditary property

Ewa Drgas-Burchardt;  Anna Fiedorowicz — 2005

Discussiones Mathematicae Graph Theory

The hereditary property of hypergraphs generated by the cost colouring notion is considered in the paper. First, we characterize all maximal graphs with respect to this property. Second, we give the generating function for the sequence describing the number of such graphs with the numbered order. Finally, we construct a maximal hypergraph for each admissible number of vertices showing some density property. All results can be applied to the problem of information storage.

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