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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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