Monika Pilśniak

A NOTE ON PACKING OF TWO COPIES OF A HYPERGRAPH

A 2-packing of a hypergraph H is a permutation ae on V (H) such that if an edge e belongs to E(H), then ae(e) does not belong to E(H). We prove that a hypergraph which does not contain neither empty edge ; nor complete edge V (H) and has at most 1 n edges is 2-packable. A 1-uniform hypergraph of order n with more than 1 n edges shows that this result cannot be improved by increasing the size of H.

Improving upper bounds for the distinguishing index

The distinguishing index of a graph G , denoted by D ʹ( G ) , is the least number of colours in an edge colouring of G not preserved by any non-trivial automorphism. We characterize all connected graphs G with D ʹ( G ) ≥ Δ ( G ) . We show that D ʹ( G ) ≤ 2 if G is a traceable graph of order at least seven, and D ʹ( G ) ≤ 3 if G is either claw-free or 3 -connected and planar. We also investigate the Nordhaus-Gaddum type relation: 2 ≤  D ʹ( G ) +  D ʹ( ‾ G ) ≤ max{Δ ( G ), Δ ( ‾ G )} + 2 and we confirm it for some classes of graphs.

Distinguishing Cartesian Products of Countable Graphs

Abstract The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.

How to personalize the vertices of a graph

Abstract If f is a proper coloring of edges in a graph G = ( V , E ) , then for each vertex v ∈ V it defines the palette of colors of v , i.e., the set of colors of edges incident with v . In 1997, Burris and Schelp stated the following problem: how many colors do we have to use if we want to distinguish all vertices by their palettes. In general, we may need much more colors than χ ′ ( G ) . In this paper we show that if we distinguish the vertices by color walks emanating from them, not just by their palettes, then the number of colors we need is very close to the chromatic index. Actually, not greater than Δ ( G ) + 1 .

Dense Arbitrarily Partitionable Graphs

Abstract A graph G of order n is called arbitrarily partitionable (AP for short) if, for every sequence (n1, . . . , nk) of positive integers with n1 + ⋯ + nk = n, there exists a partition (V1, . . . , Vk) of the vertex set V (G) such that Vi induces a connected subgraph of order ni for i = 1, . . . , k. In this paper we show that every connected graph G of order n ≥ 22 and with ‖G‖ > (n−42)+12

The distinguishing index of the Cartesian product of finite graphs

||G||\; > \;\left( {\matrix{{n - 4} \cr 2 \cr } } \right) + 12

Distinguishing graphs by total colourings

edges is AP or belongs to few classes of exceptional graphs.

On cyclically embeddable (n,n)-graphs

The distinguishing index D ʹ( G ) of a graph G is the least natural number d such that G has an edge colouring with d colours that is only preserved by the identity automorphism. In this paper we investigate the distinguishing index of the Cartesian product of connected finite graphs. We prove that for every k  ≥ 2 , the k -th Cartesian power of a connected graph G has distinguishing index equal 2 , with the only exception D ʹ( K 2 2 ) = 3 . We also prove that if G and H are connected graphs that satisfy the relation 2 ≤ ∣ G ∣ ≤ ∣ H ∣ ≤ 2^∣ G ∣(2^∣∣ G ∣∣ − 1) − ∣ G ∣ + 2 , then D ʹ( G □  H ) ≤ 2 unless G □  H  =  K 2 2 .

Edge motion and the distinguishing index

We introduce the total distinguishing number D ʺ( G ) of a graph G as the least number d such that G has a total colouring (not necessarily proper) with d colours that is only preserved by the trivial automorphism. This is an analog to the notion of the distinguishing number D ( G ) , and the distinguishing index D ʹ( G ) , which are defined for colourings of vertices and edges, respectively. We obtain a general sharp upper bound: D ʺ( G ) ≤ ⌈√Δ ( G )⌉ for every connected graph G . We also introduce the total distinguishing chromatic number χ ʺ D ( G ) similarly defined for proper total colourings of a graph G . We prove that χ ʺ D ( G ) ≤  χ ʺ( G ) + 1 for every connected graph G with the total chromatic number χ ʺ( G ) . Moreover, if χ ʺ( G ) ≥ Δ ( G ) + 2 , then χ ʺ D ( G ) =  χ ʺ( G ) . We prove these results by setting sharp upper bounds for the minimal number of colours in a proper total colouring such that each vertex has a distinct set of colour walks emanating from it.

Precise bounds for the distinguishing index of the Cartesian product

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.