Agnieszka Görlich

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.

A lower bound on the size of (H;1)-vertex stable graphs

Abstract A graph G is called ( H ; k ) -vertex stable if G contains a subgraph isomorphic to H even after removing any k of its vertices. By stab ( H ; k ) we denote the minimum size among the sizes of all ( H ; k ) -vertex stable graphs. In this paper we present a general result concerning ( H ; 1 ) -vertex stable graphs. Namely, for an arbitrary graph H we give a lower bound for stab ( H ; 1 ) in terms of the order, connectivity and minimum degree of H . The bound is nearly sharp.

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.

Decomposition of complete bipartite graphs into open trails

Abstract It has been showed in [Horňak, M., and M. Woźniak, Decomposition of complete bipartite even graphs into closed trails, Czechoslovak Mathematical Journal 128 (2003) 127-134] that any bipartite graph K a , b , where a, b are even is decomposable into closed trails of prescribed even lengths. In this article we consider the corresponding question for open trails. We show that for even a and b any complete bipartite graph K a , b is decomposable into open trails of arbitrarily lengths (less than ab) whenever these lengths sum up to the size of the graph K a , b . Let K a , a ′ : = K a , a − I a for any 1-factor I a . We also prove that for odd a K a , a ′ can be decomposed in a similar manner.

Sparse graphs of girth at least five are packable

A graph is packable if it is a subgraph of its complement. The following statement was conjectured by Faudree, Rousseau, Schelp and Schuster in 1981: every non-star graph G with girth at least 5 is packable.The conjecture was proved by Faudree et?al. with the additional condition that G has at most 6 5 n - 2 edges. In this paper, for each integer k ? 3 , we prove that every non-star graph with girth at least 5 and at most 2 k - 1 k n - α k ( n ) edges is packable, where α k ( n ) is o ( n ) for every k . This implies that the conjecture is true for sufficiently large planar graphs.

On cyclically embeddable (n,n)-graphs

An embedding of a simple graph G into its complement G 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.

Cordial labeling of hypertrees

Abstract Let H = ( V , E ) be a hypergraph with vertex set V = { v 1 , v 2 , … , v n } and edge set E = { e 1 , e 2 , … , e m } . A vertex labeling c : V → N induces an edge labeling c ∗ : E → N by the rule c ∗ ( e i ) = ∑ v j ∈ e i c ( v j ) . For integers k ≥ 2 we study the existence of labelings satisfying the following condition: every residue class modulo k occurs exactly ⌊ n / k ⌋ or ⌈ n / k ⌉ times in the sequence c ( v 1 ) , c ( v 2 ) , … , c ( v n ) and exactly ⌊ m / k ⌋ or ⌈ m / k ⌉ times in the sequence c ∗ ( e 1 ) , c ∗ ( e 2 ) , … , c ∗ ( e m ) . Hypergraph H is called k -cordial if it admits a labeling with these properties. Hovey [M. Hovey, A-cordial graphs, Discrete Math. 93 (1991) 183–194] raised the conjecture (still open for k > 5 ) that if H is a tree graph, then it is k -cordial for every k . Here we investigate the analogous problem for hypertrees (connected hypergraphs without cycles) and present various sufficient conditions on H to be k -cordial. From our theorems it follows that every k -uniform hypertree is k -cordial, and every hypertree with n or m odd is 2-cordial. Both of these results generalize Cahit’s theorem [I. Cahit, Cordial graphs: a weaker version of graceful and harmonious graphs, Ars Combin. 23 (1987) 201–207] which states that every tree graph is 2-cordial. We also prove that every uniform hyperpath is k -cordial for every k .

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].

A Note on a Packing Problem in Transitive Tournaments

Let TTn be a transitive tournament on n vertices. We show that for any directed acyclic graph G of order n and of size not greater than two directed graphs isomorphic to G are arc disjoint subgraphs of TTn. Moreover, this bound is generally the best possible.

Constant sum partition of sets of integers and distance magic graphs

Abstract 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 families of distance magic graphs.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 A, A, . . . , A, B, B, . . . , B such that |A| = m and |B| = 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 families of distance magic graphs.