Abstract
Let λ K v be the complete multigraph, G a finite simple graph. A G-design of λ K v is denoted by GD(v,G,λ). The crown graph Q n is obtained by joining single pendant edge to each vertex of an n-cycle. We give new constructions for Q n -designs. Let v and λ be two positive integers. For n=4, 6, 8 and λ≥1, there exists a GD(v,Q n ,λ) if and only if either (1) v>2n and λ v(v−1)≡0 (mod 4n), or (2) v=2n and λ≡0 (mod 4). Let n≥4 be even. Then (1) there exists a GD(2n,Q n ,λ) if and only if λ≡0 (mod 4). (2) There exists a GD(2n+1,Q n ,λ) when λ≡0 (mod 4).
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Liang, Z., Guo, J. Decomposition of complete multigraphs into crown graphs. J. Appl. Math. Comput. 32, 507–517 (2010). https://doi.org/10.1007/s12190-009-0267-0
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DOI: https://doi.org/10.1007/s12190-009-0267-0