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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 \(\varepsilon= \min_{1\leq i\leq \ell} | E(H_ i) |\).

MSC:
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
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