Mariusz Woźniak

On the Neighbour-Distinguishing Index of a Graph

A proper edge colouring of a graph G is neighbour-distinguishing provided that it distinguishes adjacent vertices by sets of colours of their incident edges. It is proved that for any planar bipartite graph G with Δ(G)≥12 there is a neighbour-distinguishing edge colouring of G using at most Δ(G)+1 colours. Colourings distinguishing pairs of vertices that satisfy other requirements are also considered.

Decomposition of Complete Bipartite Even Graphs into Closed Trails

We prove that any complete bipartite graph Ka,b, where a, b are even integers, can be decomposed into closed trails with prescribed even lengths.

Embedding graphs of small size

Abstract An embedding of a graph G ( V, E ) into its complement is a permutation ω on V(G) such that if an edge xy belongs to E(G) then ω( x )ω( y ) does not belong to E(G) . The fact that every graph G of order n and size less than or equal to n −2 is embeddable is well known and has been improved in many ways. We present these improvements which give more information about embeddings than just the existence. The new results (Theorems 1.8, 1.10, 1.12 and 1.13) concern the existence of embeddings σ with restrictions on the cycle length of σ considered as a permutation.

A note on the vertex-distinguishing proper coloring of graphs with large minimum degree

Abstract We prove that the number of colors required to properly color the edges of a graph of order n and δ(G)>n/3 in such a way that any two vertices are incident with different sets of colors is at most Δ(G)+5.

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.

Packing of graphs and permutations—a survey

Abstract An embedding of a graph G (into its complement Ḡ) 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). If there exists an embedding of G we say that G is embeddable or that there is a packing of two copies of the graph G into complete graph Kn. In this paper we discuss a variety of results, some quite recent, concerning the relationships between the embeddings of graphs in their complements and the structure of the embedding permutations.

Triple placement of graphs

A graphG of ordern is said to be 3-placeable if there are three edge disjoint copies ofG inKn. In the paper we prove that every graph of ordern and size at mostn−2 is 3-placeable unless isomorphic either toK3 υ 2K1 or toK4 υ 4K1.

Edge-disjoint placement of three trees

Abstract We present complete results concerning edge-disjoint placement of three trees of orderninto the complete graphkn.

Packing three trees

Abstract We present some results concerning edge-disjoint placement of two or three copies of a tree, as well as a theorem about the packing of three trees into the complete graph K n .

On the packing of three graphs

Abstract We prove that three graphs of order n and size less than or equal to n −3 can be packed (edge-disjointly) into the complete graph K n .