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Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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On generating snarks

100%
Chisala B.
Discussiones Mathematicae Graph Theory
|
1998
|
tom 18
|
nr 2
147-158
EN
We discuss the construction of snarks (that is, cyclically 4-edge connected cubic graphs of girth at least five which are not 3-edge colourable) by using what we call colourable snark units and a welding process.
2
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Fibonacci and Telephone Numbers in Extremal Trees

100%
Bednarz U., Włoch I.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 38
|
nr 1
121-133
EN
In this paper we shall show applications of the Fibonacci numbers in edge-coloured trees. In particular we determine the successive extremal graphs in the class of trees with respect to the number of (A, 2B)-edge colourings. We show connections between these numbers and Fibonacci numbers as well as the telephone numbers.
3
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Graphs with rainbow connection number two

100%
Kemnitz A., Schiermeyer I.
Discussiones Mathematicae Graph Theory
|
2011
|
tom 31
|
nr 2
313-320
EN
An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. In this paper we prove that rc(G) = 2 for every connected graph G of order n and size m, where $\binom{n-1}{2} + 1 ≤ m ≤ \binom{n}{2} - 1$. We also characterize graphs with rainbow connection number two and large clique number.
4
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Forbidden Structures for Planar Perfect Consecutively Colourable Graphs

88%
Borowiecka-Olszewska M., Drgas-Burchardt E.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 2
315-336
EN
A consecutive colouring of a graph is a proper edge colouring with posi- tive integers in which the colours of edges incident with each vertex form an interval of integers. The idea of this colouring was introduced in 1987 by Asratian and Kamalian under the name of interval colouring. Sevast- janov showed that the corresponding decision problem is NP-complete even restricted to the class of bipartite graphs. We focus our attention on the class of consecutively colourable graphs whose all induced subgraphs are consecutively colourable, too. We call elements of this class perfect consecutively colourable to emphasise the conceptual similarity to perfect graphs. Obviously, the class of perfect consecutively colourable graphs is induced hereditary, so it can be characterized by the family of induced forbidden graphs. In this work we give a necessary and sufficient conditions that must be satisfied by the generalized Sevastjanov rosette to be an induced forbid- den graph for the class of perfect consecutively colourable graphs. Along the way, we show the exact values of the deficiency of all generalized Sevastjanov rosettes, which improves the earlier known estimating result. It should be mentioned that the deficiency of a graph measures its closeness to the class of consecutively colourable graphs. We motivate the investigation of graphs considered here by showing their connection to the class of planar perfect consecutively colourable graphs.
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