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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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##### References:
 [1] Alspach, B.; Gavlas, H., Cycle decompositions of $$K$$_{$$n$$} and $$K$$_{$$n$$} − $$I$$, J. Combin. Theory Ser. B, 81, 77-99, (2001) · Zbl 1023.05112 [2] Andersen, L.D.; Hilton, A.J.W., Generalized Latin rectangles I: construction and decomposition, Discrete Math., 31, 125-152, (1980) · Zbl 0443.05019 [3] Andersen, L.D.; Hilton, A.J.W., Generalized Latin rectangles II: embedding, Discrete Math., 31, 235-260, (1980) · Zbl 0476.05018 [4] Andersen, L.D.; Rodger, C.A., Decompositions of complete graphs: embedding partial edge-colourings and the method of amalgamations, Surveys in Combinatorics Lond Math Soc Lect Note Ser, 307, 7-41, (2003) · Zbl 1028.05083 [5] Bahmanian, M.A., Rodger, C.A.: Multiply balanced edge colorings of multigraphs. J. Graph Theory. doi:10.1002/jgt.20617 · Zbl 1244.05085 [6] Bose, R.C.; Shimamoto, T., Classification and analysis of partially balanced incomplete block designs with two associate classes, J. Am. Statist. Assoc., 47, 151-184, (1952) · Zbl 0048.11603 [7] Fu, H.L.; Rodger, C.A., Group divisible designs with two associate classes: $$n$$ = 2 or $$m$$ = 2, J. Comb. Theory Ser. A, 83, 94-117, (1998) · Zbl 0911.05024 [8] Fu, H.L.; Rodger, C.A., 4-cycle group-divisible designs with two associate classes, Comb. Probab. Comput., 10, 317-343, (2001) · Zbl 0992.05030 [9] Fu, H.L.; Rodger, C.A.; Sarvate, D.G., The existence of group divisible designs with first and second associates, having block size 3, Ars Comb., 54, 33-50, (2000) · Zbl 0993.05018 [10] Hall, M., An existence theorem for Latin squares, Bull. Am. Math. Soc., 51, 387-388, (1945) · Zbl 0060.02801 [11] Hilton, A.J.W., Hamiltonian decompositions of complete graphs, J. Comb. Theory B, 36, 125-134, (1984) · Zbl 0542.05044 [12] Hilton, A.J.W.; Johnson, M.; Rodger, C.A.; Wantland, E.B., Amalgamations of connected $$k$$-factorizations, J. Comb. Theory B, 88, 267-279, (2003) · Zbl 1033.05084 [13] Hilton, A.J.W.; Rodger, C.A., Hamiltonian decompositions of complete regular $$s$$-partite graphs, Discrete Math., 58, 63-78, (1986) · Zbl 0593.05047 [14] Laskar, R.; Auerbach, B., On decomposition of $$r$$-partite graphs into edge-disjoint Hamilton circuits, Discrete Math., 14, 265-268, (1976) · Zbl 0322.05128 [15] Lindner, C.C.; Rodger, C.A., Generalized embedding theorems for partial Latin squares, Bull. Inst. Comb. Appl., 5, 81-99, (1992) · Zbl 0829.05014 [16] Lucas E.: Récréations Mathématiques. vol. 2. Gauthiers Villars, Paris (1883) [17] Nash-Williams, C.St.J.A., Amalgamations of almost regular edge-colourings of simple graphs, J. Comb. Theory B, 43, 322-342, (1987) · Zbl 0654.05031 [18] Rodger, C.A.; Wantland, E.B., Embedding edge-colorings into 2-edge-connected $$k$$-factorizations of $$K$$_{kn+1}, J. Graph Theory, 19, 169-185, (1995) · Zbl 0815.05050 [19] S̆ajna, M.: Cycle decompositions of $$K$$_{$$n$$} and $$K$$_{$$n$$}−$$I$$. PhD Thesis Simon Fraser University (1999) · Zbl 1028.05083 [20] S̆ajna, M., Cycle decompositions III: complete graphs and fixed length cycles, J. Comb. Des., 10, 27-78, (2002) · Zbl 1033.05078 [21] Tarsi, M., On the decomposition of a graph into stars, Discrete Math., 36, 299-304, (1981) · Zbl 0467.05054 [22] Tarsi, M., Decomposition of a complete multigraph into simple paths: nonbalanced handcuffed designs, J.Comb. Theory Ser. A, 34, 60-70, (1983) · Zbl 0511.05024
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