Antoni Marczyk

Arbitrarily vertex decomposable caterpillars with four or five leaves

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (a1, . . . , ak) of positive integers such that a1+. . .+ak = n there exists a partition (V1, . . . , Vk) of the vertex set of G such that for each i ∈ {1, . . . , k}, Vi induces a connected subgraph of G on ai vertices. D. Barth and H. Fournier showed that if a tree T is arbitrarily vertex decomposable, then T has maximum degree at most 4. In this paper we give a complete characterization of arbitrarily vertex decomposable caterpillars with four leaves. We also describe two families of 292 S. Cichacz, A. Görlich, A. Marczyk, J. PrzybyÃlo and ... arbitrarily vertex decomposable trees with maximum degree three or four.

Dense Arbitrarily Vertex Decomposable Graphs

A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n1, . . . , nk) of positive integers such that n1 + · · · + nk = n there exists a partition (V1, . . . , Vk) of the vertex set of G such that for each

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

{i \in \{1,\ldots,k\}}

A note on pancyclism of highly connected graphs

, Vi induces a connected subgraph of G on ni vertices. The main result of the paper reads as follows. Suppose that G is a connected graph on n ≥ 20 vertices that admits a perfect matching or a matching omitting exactly one vertex. If the degree sum of any pair of nonadjacent vertices is at least n − 5, then G is arbitrarily vertex decomposable. We also describe 2-connected arbitrarily vertex decomposable graphs that satisfy a similar degree sum condition.

Chvátal-Erdos condition and pancyclism

The purpose of this note is to study the problem of cyclability under the condition called regional Ore’s condition. As a consequence, we get the hamiltonicity of a graph G for which Ore’s condition holds in each of k vertex subsets partitioning V(G) separately (regionally), provided that the graph is k connected.

On the Set of Cycle Lengths in a Hamiltonian Graph with a Given Maximum Degree

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.

A note on a new condition implying pancyclism

We study the set of cycle lengths in a hamiltonian graph G of order n with two fixed and nonadjacent vertices x, y whose degree sum satisfies d(x)+d(y)>=n+z, where z>=1. We prove that this set contains all integers between 3 and 4n+4z+3219. This improves and generalizes some results of Faudree et al. (Discuss. Math. Graph Theory 16 (1996) 27) and Schelten and Schiermeyer (Discrete Appl. Math. 79 (1997) 201). We also show that if z>=518n then G contains a cycle of length p for every p satisfying 3=

On pancyclism of hamiltonian 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).