Mariusz Woniak

On-line arbitrarily vertex decomposable trees

A tree T is arbitrarily vertex decomposable if for any sequence @t of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by @t. An on-line version of the problem of characterizing arbitrarily vertex decomposable trees is completely solved here.

On arbitrarily vertex decomposable trees

A tree T is arbitrarily vertex decomposable if for any sequence @t of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by @t; from a result by Barth and Fournier it follows that @D(T)=<4. A necessary and a sufficient condition for being an arbitrarily vertex decomposable star-like tree have been exhibited. The conditions seem to be very close to each other.

On-line arbitrarily vertex decomposable suns

We give a complete characterization of on-line arbitrarily vertex decomposable graphs in the family of unicycle graphs called suns. A sun is a graph with maximum degree three, such that deleting vertices of degree one results in a cycle. This result has already been used in another paper to prove some Ore-type conditions for on-line arbitrarily decomposable graphs.

A generalization of Dirac's theorem on cycles through k vertices in k-connected graphs

Let X be a subset of the vertex set of a graph G. We denote by @k(X) the smallest number of vertices separating two vertices of X if X does not induce a complete subgraph of G, otherwise we put @k(X)=|X|-1 if |X|>=2 and @k(X)=1 if |X|=1. We prove that if @k(X)>=2 then every set of at most @k(X) vertices of X is contained in a cycle of G. Thus, we generalize a similar result of Dirac. Applying this theorem we improve our previous result involving an Ore-type condition and give another proof of a slightly improved version of a theorem of Broersma et al.

On the structure of arbitrarily partitionable graphs with given connectivity

A graph G=(V,E) is arbitrarily partitionable if for any sequence @t of positive integers adding up to |V|, there is a sequence of vertex-disjoint subsets of V whose orders are given by @t, and which induce connected subgraphs. Such a graph models, e.g., a computer network which may be arbitrarily partitioned into connected subnetworks. In this paper we study the structure of such graphs and prove that unlike in some related problems, arbitrarily partitionable graphs may have arbitrarily many components after removing a cutset of a given size >=2. The sizes of these components grow exponentially, though.

Fixed-point-free embeddings of digraphs with small size

We prove that every digraph of order n and size at most 74n-81 is embeddable in its complement. Moreover, for such digraphs there are embeddings without fixed points.

A note on pancyclism of highly connected graphs

Jackson and Ordaz conjectured that if the stability number @a(G) of a graph G is no greater than its connectivity @k(G) then G is pancyclic. Applying Ramseys theorem we prove the pancyclicity of every graph G with @k(G) sufficiently large with respect to @a(G).

On self-complementary supergraphs of (n,n)-graphs

We prove that, with one exception, each (n,n)-graph G that is embeddable in its complement has a self-complementary supergraph of order n.

Equitable neighbour-sum-distinguishing edge and total colourings

With any (not necessarily proper) edge k-colouring :E(G){1,,k} of a graph G, one can associate a vertex colouring given by (v)=ev(e). A neighbour-sum-distinguishing edge k-colouring is an edge colouring whose associated vertex colouring is proper. The neighbour-sum-distinguishing index of a graph G is then the smallest k for which G admits a neighbour-sum-distinguishing edge k-colouring. These notions naturally extend to total colourings of graphs that assign colours to both vertices and edges.We study in this paper equitable neighbour-sum-distinguishing edge colourings and total colourings, that is colourings for which the number of elements in any two colour classes of differ by at most one. We determine the equitable neighbour-sum-distinguishing index of complete graphs, complete bipartite graphs and forests, and the equitable neighbour-sum-distinguishing total chromatic number of complete graphs and bipartite graphs.

A note on decompositions of transitive tournaments

For any positive integer n, we determine all connected digraphs G of size at most four, such that a transitive tournament of order n is G-decomposable. Among others, these results disprove a generalization of a theorem of Sali and Simonyi [Orientations of self-complementary graphs and the relation of Sperner and Shannon capacities, European J. Combin. 20 (1999), 93-99].