For any integer k ≥ 3 , we define sunlet graph of order 2k, denoted by L 2k , as the graph consisting of a cycle of length k together with k pendant vertices, each adjacent to exactly one vertex of the cycle. In this paper, we give necessary and sufficient conditions for the existence of L8-decomposition of tensor product and wreath product of complete graphs.
Let n ≥ 3 and λ ≥ 1 be integers. Let λK n denote the complete multigraph with edge-multiplicity λ. In this paper, we show that there exists a symmetric Hamilton cycle decomposition of λK 2m for all even λ ≥ 2 and m ≥ 2. Also we show that there exists a symmetric Hamilton cycle decomposition of λK 2m − F for all odd λ ≥ 3 and m ≥ 2. In fact, our results together with the earlier results (by Walecki and Brualdi and Schroeder) completely settle the existence of symmetric Hamilton cycle decomposition of λK n (respectively, λK n − F , where F is a 1-factor of λK n ) which exist if and only if λ(n − 1) is even (respectively, λ(n − 1) is odd), except the non-existence cases n ≡ 0 or 6 (mod 8) when λ = 1.
Abstract.The following question is raised by Alspach, Bermond and Sotteau: If G1 has a decomposition into hamilton cycles and a 1-factor, and G2 has a hamilton cycle decmposition (HCD), does their wreath product G1 * G2 admit a hamilton cycle decomposition? In this paper the above question is answered with an additional condition on G~. Further it is shown that some product graphs can be decomposed into cycles of uniform length, that is, the edge sets of the graphs can be partitioned into cycles of length k, for some suitable k.
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