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Embedding edge-colorings into 2-edge-connected \(k\)-factorizations of \(K_{kn+1}\). (English) Zbl 0815.05050
Using the method of amalgamations introduced by A. J. W. Hilton, the authors provide another proof of a theorem of C. St. J. A. Nash-Williams. To be specific, we say that an edge-coloring of the complete graph \(K_ r\), with colors \(c_ 1, c_ 2, \dots, c_ \ell\) and \(H_ i\) the subgraph induced by the edges of color \(c_ i\), is \(k\)-admissible if for all \(1\leq i\leq \ell\):
1. \(d_{H_ i} (v)\leq k\), for every \(v\in V(H_ i)\),
2. for every component \(C\) of \(H_ i\), \(\sum_{v\in V(C)} d_{H_ i} (v)\leq k| V(C)| -2\), and
3. if \(e\) is a cut edge of a component \(C\) of \(H_ i\), with \(C_ 1\) and \(C_ 2\) being the components of \(C-e\), then there exists \(v_ 1\in V(C_ 1)\) and \(v_ 2\in V(C_ 2)\) so that \(d_{H_ i} (v_ 1) <k\) and \(d_{H_ i} (v_ 2) <k\).
The main theorem then states that for \(r<k\ell +1\), an \(\ell\)-edge- coloring of \(K_ r\) can be embedded into a 2-edge-connected \(k\)-factor decomposition of \(K_{k \ell+1}\) if and only if the \(\ell\)-edge-coloring is \(k\)-admissible, \(\ell\) is odd if \(k\) is odd, and \(k \ell+1 \geq 2r-2 \varepsilon/k\), where \(\varepsilon= \min_{1\leq i\leq \ell} | E(H_ i) |\).

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C10 Planar graphs; geometric and topological aspects of graph theory
05C15 Coloring of graphs and hypergraphs
Full Text: DOI
[1] Andersen, Discrete Math. 31 pp 125– (1980)
[2] Andersen, Discrete Math. 31 pp 235– (1980)
[3] On the factorization of the complete uniform hypergraph. Infinite and Finite Sets. North-Holland, Amsterdam (1975) 91–108.
[4] Baranyai, J. Combinat. Theory B 26 pp 276– (1979)
[5] and , Graph Theory with Applications. Macmillan, London (1976). · Zbl 1226.05083 · doi:10.1007/978-1-349-03521-2
[6] Doyen, Discrete Math. 5 pp 229– (1973)
[7] Hanani, Can. J. Math. 12 pp 145– (1960) · Zbl 0092.01202 · doi:10.4153/CJM-1960-013-3
[8] Hilton, J. Combinat. Theory B 36 pp 125– (1984)
[9] Hilton, J. Combinat. Theory A 56 pp 109– (1991)
[10] Hilton, Discrete Math. 48 pp 63– (1986)
[11] Hoffman, J. Graph Theory 13 pp 417– (1989)
[12] Kirkman, Camb. Dublin Math. J. 2 pp 191– (1847)
[13] and , Decompositions into cycles II: cycle systems. Contemporary Design Theory: A Collection of Surveys. Wiley, New York (1992).
[14] Lindner, Discrete Math. 77 pp 191– (1989)
[15] Lindner, Discrete Math. 117 pp 151– (1993)
[16] Lindner, J. Combinat. Designs. 1 pp 113– (1993)
[17] Nash-Williams, J. London Math. Soc.
[18] Nash-Williams, Proceedings of the 10th British Combinatorics Conference, Surveys in Combinatorics (1985)
[19] Nash-Williams, J. Combinat. Theory B 43 pp 322– (1987)
[20] Tarsi, Discrete Math. 26 pp 273– (1979)
[21] Tarsi, J. Combinat. Theory A 34 pp 60– (1983)
[22] deWerra, Rev. Fran. Inf. Rech. Oper. 5 pp 3– (1971)
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