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Note on cyclic decompositions of complete bipartite graphs into cubes

100%

Fronček D.

Discussiones Mathematicae Graph Theory

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1999

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tom 19

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nr 2

219-227

EN

So far, the smallest complete bipartite graph which was known to have a cyclic decomposition into cubes $Q_d$ of a given dimension d was $K_{d2^{d-1}, d2^{d-2}}$. We improve this result and show that also $K_{d2^{d-2}, d2^{d-2}}$ allows a cyclic decomposition into $Q_d$. We also present a cyclic factorization of $K_{8,8}$ into Q₄.

2

2-halvable complete 4-partite graphs

100%

Fronček D.

Discussiones Mathematicae Graph Theory

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1998

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tom 18

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nr 2

233-242

EN

A complete 4-partite graph $K_{m₁,m₂,m₃,m₄}$ is called d-halvable if it can be decomposed into two isomorphic factors of diameter d. In the class of graphs $K_{m₁,m₂,m₃,m₄}$ with at most one odd part all d-halvable graphs are known. In the class of biregular graphs $K_{m₁,m₂,m₃,m₄}$ with four odd parts (i.e., the graphs $K_{m,m,m,n}$ and $K_{m,m,n,n}$) all d-halvable graphs are known as well, except for the graphs $K_{m,m,n,n}$ when d = 2 and n ≠ m. We prove that such graphs are 2-halvable iff n,m ≥ 3. We also determine a new class of non-halvable graphs $K_{m₁,m₂,m₃,m₄}$ with three or four different odd parts.

3

Product rosy labeling of graphs

100%

Fronček D.

Discussiones Mathematicae Graph Theory

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2008

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tom 28

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nr 3

431-439

EN

In this paper we describe a natural extension of the well-known ρ-labeling of graphs (also known as rosy labeling). The labeling, called product rosy labeling, labels vertices with elements of products of additive groups. We illustrate the usefulness of this labeling by presenting a recursive construction of infinite families of trees decomposing complete graphs.

4

On traceability and 2-factors in claw-free graphs

51%

Fronček D., Ryjáček Z., Skupień Z.

Discussiones Mathematicae Graph Theory

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2004

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tom 24

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nr 1

55-71

EN

If G is a claw-free graph of sufficiently large order n, satisfying a degree condition σₖ > n + k² - 4k + 7 (where k is an arbitrary constant), then G has a 2-factor with at most k - 1 components. As a second main result, we present classes of graphs 𝓒₁,...,𝓒₈ such that every sufficiently large connected claw-free graph satisfying degree condition σ₆(k) > n + 19 (or, as a corollary, δ(G) > (n+19)/6) either belongs to $⋃ ⁸_{i=1} 𝓒_i$ or is traceable.

Ograniczanie wyników

4 Discussiones Mathematicae Graph Theory

4 Fronček D.

1 Ryjáček Z.

1 Skupień Z.

1 2008

1 2004

1 1999

1 1998

Note on cyclic decompositions of complete bipartite graphs into cubes

2-halvable complete 4-partite graphs

Product rosy labeling of graphs

On traceability and 2-factors in claw-free graphs