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On chromaticity of graphs

100%
Łazuka E.
Discussiones Mathematicae Graph Theory
|
1995
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tom 15
|
nr 1
19-31
EN
In this paper we obtain the explicit formulas for chromatic polynomials of cacti. From the results relating to cacti we deduce the analogous formulas for the chromatic polynomials of n-gon-trees. Besides, we characterize unicyclic graphs by their chromatic polynomials. We also show that the so-called clique-forest-like graphs are chromatically equivalent.
2
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Chromatic polynomials of hypergraphs

63%
Borowiecki M., Łazuka E.
Discussiones Mathematicae Graph Theory
|
2000
|
tom 20
|
nr 2
293-301
EN
In this paper we present some hypergraphs which are chromatically characterized by their chromatic polynomials. It occurs that these hypergraphs are chromatically unique. Moreover we give some equalities for the chromatic polynomials of hypergraphs generalizing known results for graphs and hypergraphs of Read and Dohmen.
3
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Colouring of cycles in the de Bruijn graphs

63%
Łazuka E., Żurawiecki J.
Discussiones Mathematicae Graph Theory
|
2000
|
tom 20
|
nr 1
5-21
EN
We show that the problem of finding the family of all so called the locally reducible factors in the binary de Bruijn graph of order k is equivalent to the problem of finding all colourings of edges in the binary de Bruijn graph of order k-1, where each vertex belongs to exactly two cycles of different colours. In this paper we define and study such colouring for the greater class of the de Bruijn graphs in order to define a class of so called regular factors, which is not so difficult to construct. Next we prove that each locally reducible factor of the binary de Bruijn graph is a subgraph of a certain regular factor in the m-ary de Bruijn graph.
4
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Rainbow numbers for small stars with one edge added

63%
Gorgol I., Łazuka E.
Discussiones Mathematicae Graph Theory
|
2010
|
tom 30
|
nr 4
555-562
EN
A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. For a graph H and a positive integer n, the anti-Ramsey number f(n,H) is the maximum number of colors in an edge-coloring of Kₙ with no rainbow copy of H. The rainbow number rb(n,H) is the minimum number of colors such that any edge-coloring of Kₙ with rb(n,H) number of colors contains a rainbow copy of H. Certainly rb(n,H) = f(n,H) + 1. Anti-Ramsey numbers were introduced by Erdös et al. [5] and studied in numerous papers. We show that $rb(n,K_{1,4} + e) = n + 2$ in all nontrivial cases.
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