Mieczysław Borowiecki

A survey of hereditary properties of graphs

In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure, is emphasized. Many related topics, including questions on the complexity of related problems, are investigated.

Chromatic polynomials of hypergraphs

In this paper we present some hypergraphs which are chromatically characterized by their chromatic polynomials. It occurs that these hypergraphs are chromatically unique. Moreover we give some equalities for the chromatic polynomials of hypergraphs generalizing known results for graphs and hypergraphs of Read and Dohmen.

GENERALIZED LIST COLOURINGS OF GRAPHS

1. Introduction and NotationWe consider finite undirected graphs without loops and multiple edges. Thevertex set of a graph G is denoted by V(G) and the edge set by E(G). Thenotation H ⊆ G means that H is a subgraph of G. The vertex induced (wewill say briefly: induced) subgraph H of G is denoted by H ≤G. We say thatG contains H whenever G contains a subgraph isomorphic to H. In general,we follow the notation and terminology of [5].Let I denote the set of all mutually nonisomorphic graphs. If P is anonempty subset of I, then P will also denote the property that a graph is

Matching cutsets in graphs of diameter 2

We say that a graph has a matching cutset if its vertices can be coloured in red and blue in such a way that there exists at least one vertex coloured in red and at least one vertex coloured in blue, and every vertex has at most one neighbour coloured in the opposite colour. In this paper we study the algorithmic complexity of a problem of recognizing graphs which possess a matching cutset. In particular we present a polynomial-time algorithm which solves this problem for graphs of diameter two.

Minimal reducible bounds for planar graphs

Abstract For properties of graphs P 1 and P 2 a vertex ( P 1 , P 2 ) -partition of a graph G is a partition (V 1 ,V 2 ) of V(G) such that each subgraph G[V i ] induced by V i has property P i , i=1,2 . The class of all vertex ( P 1 , P 2 ) -partitionable graphs is denoted by P 1 ∘ P 2 . An additive hereditary property R is reducible if there exist additive hereditary properties P 1 and P 2 such that R = P 1 ∘ P 2 , otherwise it is irreducible. For a given property P a reducible property R is called a minimal reducible bound for P if P ⊆ R and there is no reducible property R ′ satisfying P ⊆ R ′⊂ R . In this paper we give a survey of known reducible bounds and we prove some new minimal reducible bounds for important classes of planar graphs. The connection between our results and Barnettes conjecture is also presented.

Coloring chip configurations on graphs and digraphs

Let D be a simple directed graph. Suppose that each edge of D is assigned with some number of chips. For a vertex v of D, let q^+(v) and q^-(v) be the total number of chips lying on the arcs outgoing form v and incoming to v, respectively. Let q(v)=q^+(v)-q^-(v). We prove that there is always a chip arrangement, with one or two chips per edge, such that q(v) is a proper coloring of D. We also show that every undirected graph G can be oriented so that adjacent vertices have different balanced degrees (or even different in-degrees). The arguments are based on peculiar chip shifting operation which provides efficient algorithms for obtaining the desired chip configurations. We also investigate modular versions of these problems. We prove that every k-colorable digraph has a coloring chip configuration modulo k or k+1.

Generalized total colorings of graphs

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be additive hereditary properties of graphs. A (P ,Q)-total coloring ∗Research supported in part by Slovak VEGA Grant 2/0194/10. 210 M. Borowiecki, A. Kemnitz, M. Marangio and P. Mihok of a simple graphG is a coloring of the vertices V (G) and edgesE(G) of G such that for each color i the vertices colored by i induce a subgraph of property P , the edges colored by i induce a subgraph of property Q and incident vertices and edges obtain different colors. In this paper we present some general basic results on (P ,Q)-total colorings. We determine the (P ,Q)-total chromatic number of paths and cycles and, for specific properties, of complete graphs. Moreover, we prove a compactness theorem for (P ,Q)-total colorings.

Generalized independence and domination in graphs

Abstract The purpose of this paper is to introduce various concepts of P -domination, which generalize and unify different well-known kinds of domination in graphs. We generalize a result of Lovasz concerning the existence of a partition of a set of vertices of G into independent subsets and a result of Favaron concerning a property of S k -dominating sets. Gallai-type equalities for the strong P -domination number are proved, which generalize Nieminens result.

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.

Weakly P-saturated graphs

For a hereditary property P let kP(G) denote the number of forbidden subgraphs contained in G. A graph G is said to be weakly Psaturated, if G has the property P and there is a sequence of edges of G, say e1, e2, . . . , el, such that the chain of graphs G = G0 ⊂ G0+e1 ⊂ G1 + e2 ⊂ . . . ⊂ Gl−1 + el = Gl = Kn (Gi+1 = Gi + ei+1) has the following property: kP(Gi+1) > kP(Gi), 0 ≤ i ≤ l − 1. In this paper we shall investigate some properties of weakly saturated graphs. We will find upper bound for the minimum number of edges of weakly Dk-saturated graphs of order n. We shall determine the number wsat(n,P) for some hereditary properties.