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Embedding an edge-colored \(K(a^{(p)};\lambda,\mu)\) into a Hamiltonian decomposition of \(K(a^{(p+r)};\lambda,\mu)\). (English) Zbl 1268.05050
Summary: Let \(K(a^{(p)};\lambda,\mu)\) be a graph with \(p\) parts, each part having size \(a\), in which the multiplicity of each pair of vertices in the same part (in different parts) is \(\lambda\) (\(\mu\), respectively). In this paper we consider the following embedding problem: When can a graph decomposition of \(K(a^{(p)};\lambda,\mu)\) be extended to a Hamiltonian decomposition of \(K(a^{(p+r)};\lambda,\mu)\) for \(r>0\)? A general result is proved, which is then used to solve the embedding problem for all \(r\geq\frac{\lambda}{\mu a}+\frac{p-1}{a-1}\). The problem is also solved when \(r\) is as small as possible in two different senses, namely when \(r=1\) and when \(r=\frac{\lambda}{\mu a}-p+1\).

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C15 Coloring of graphs and hypergraphs
05C51 Graph designs and isomorphic decomposition
05C38 Paths and cycles
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