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Constant Sum Partition of Sets of Integers and Distance Magic Graphs

Sylwia Cichacz;  Agnieszka Gőrlich — 2018

Discussiones Mathematicae Graph Theory

Let A = {1, 2, . . . , tm+tn}. We shall say that A has the (m, n, t)-balanced constant-sum-partition property ((m, n, t)-BCSP-property) if there exists a partition of A into 2t pairwise disjoint subsets A1, A2, . . . , At, B1, B2, . . . , Bt such that |Ai| = m and |Bi| = n, and ∑a∈Ai a = ∑b∈Bj b for 1 ≤ i ≤ t and 1 ≤ j ≤ t. In this paper we give sufficient and necessary conditions for a set A to have the (m, n, t)-BCSP-property in the case when m and n are both even. We use this result to show some...

On cyclically embeddable (n,n)-graphs

Agnieszka Görlich;  Monika Pilśniak;  Mariusz Woźniak — 2003

Discussiones Mathematicae Graph Theory

An embedding of a simple graph G into its complement G̅ is a permutation σ on V(G) such that if an edge xy belongs to E(G), then σ(x)σ(y) does not belong to E(G). In this note we consider the embeddable (n,n)-graphs. We prove that with few exceptions the corresponding permutation may be chosen as cyclic one.

Arbitrarily vertex decomposable caterpillars with four or five leaves

Sylwia Cichacz;  Agnieszka Görlich;  Antoni Marczyk;  Jakub Przybyło;  Mariusz Woźniak — 2006

Discussiones Mathematicae Graph Theory

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, V i induces a connected subgraph of G on a i 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...

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