Rafał Kalinowski

A catalogue of small maximal nonhamiltonian graphs

In this paper a catalogue of all maximal nonhamiltonian graphs of orders up to 10 is provided. Special attention is paid to maximal nonhamiltonian graphs with non-positive scattering number since all remaining ones (with scattering number 1) are fully characterized and counted by the third author. We also give a sketch of the method used to produce the catalogue.

Periodic Points of Continuous Mappings of Trees

Let f be a continuous map of a tree T into itself with a periodic point of period n. We show that f has points of arbitrarily large periods if n is divisible by an odd number h which is larger than the number of edges of T. Moreover, we prove the validity of a considerably stronger conclusion for the case when h is prime. Our results generalise the theorem of Sarkovskii about continuous maps of the interval.

On-line arbitrarily vertex decomposable suns

We give a complete characterization of on-line arbitrarily vertex decomposable graphs in the family of unicycle graphs called suns. A sun is a graph with maximum degree three, such that deleting vertices of degree one results in a cycle. This result has already been used in another paper to prove some Ore-type conditions for on-line arbitrarily decomposable 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.

Research Problems from the Fourth Cracow Conference (Czorsztyn, 2002)

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 .

A note on decompositions of transitive tournaments

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.