Anna Fiedorowicz

On partitions of hereditary properties of graphs

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 wellknown concept of Ramsey Class of graphs. Some Q-Ramsey Classes of graphs are presented (Theorem 1 and 2). We proved that T 2, 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 T 2 we found two bounds (Theorem 4). An improvement, in some sense, of that in Theorem 5 is given.

Acyclic colorings of graphs with bounded degree

A k-coloring (not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colors i and j the subgraph induced by the edges whose endpoints have colors i and j is acyclic. We consider some generalized acyclic k-colorings, namely, we require that each color class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic 5-coloring such that each color class induces an acyclic graph with maximum degree at most 4. We prove that the problem of deciding whether a graph G has an acyclic 2-coloring in which each color class induces a graph with maximum degree at most 3 is NP-complete, even for graphs with maximum degree 5. We also give a linear-time algorithm for an acyclic t-improper coloring of any graph with maximum degree d assuming that the number of colors is large enough.

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

Abstract 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 been widely studied. In particular, the problem of acyclic colourings of graphs with bounded maximum degree has been investigated. In 2011, Kostochka and Stocker showed that any graph with maximum degree 5 can be acyclically coloured with at most 7 colours. The question, whether this bound is achieved, remains open. In this note we prove that any graph with maximum degree 5 and maximum average degree at most 4 admits an acyclic 6-colouring. We also provide examples of graphs with these properties.

Acyclic reducible bounds for outerplanar graphs

For a given graph G and a sequence P1,P2, . . . ,Pn of additive hereditary classes of graphs we define an acyclic (P1,P2, . . . ,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[Vi] ∈ Pi 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 ∈ Vi and v ∈ Vj is acyclic. A class R = P1 P2 · · · Pn is defined as the set of the graphs having an acyclic (P1,P2, . . . ,Pn)-colouring. If P ⊆ R, then we say that R is an acyclic reducible bound for P. In this paper we present acyclic reducible bounds for the class of outerplanar graphs.

Cost colourings of hypergraphs

Some aspects of the cost colouring of hypergraphs are considered in the paper. A generalisation of the well-known Brooks theorem for a cost colouring of hypergraphs is proved. Moreover, a relation between the minimal cost of a colouring of a hypergraph with a set of costs which produce an arithmetic sequence and a number of edges of this hypergraph is investigated.

Maximal Hypergraphs with Respect to the Bounded Cost Hereditary Property

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.