Abstract
In this paper, we consider the problem of decomposing the complete directed graph $K_n^*$ into cycles of given lengths. We consider general necessary conditions for a directed cycle decomposition of $K_n^*$ into $t$ cycles of lengths $m_1, m_2, \ldots, m_t$ to exist and and provide a powerful construction for creating such decompositions in the case where there is one 'large' cycle. Finally, we give a complete solution in the case when there are exactly three cycles of lengths $\alpha, \beta, \gamma \neq 2$. Somewhat surprisingly, the general necessary conditions turn out not to be sufficient in this case. In particular, when $\gamma=n$, $\alpha+\beta > n+2$ and $\alpha+\beta \equiv n$ (mod 4), $K_n^*$ is not decomposable.
Highlights
IntroductionA more general question is the existence of possibly non-uniform cycle decompositions of Kn or Kn − I
Let G be a graph and H = {H1, H2, . . . , Hr} be a collection of subgraphs of G
Theorem 16 applies when the underlying graph G formed by cycles not of length the electronic journal of combinatorics 28(1) (2021), #P1.35 two is sparse
Summary
A more general question is the existence of possibly non-uniform cycle decompositions of Kn or Kn − I It was conjectured by Alspach [1] in 1981 that the obvious necessary conditions for the existence of such a decomposition were sufficient. Bryant, Horsley, Maenhaut and Smith [4] have extended this result, finding necessary and sufficient conditions for the existence of a cycle decomposition of the complete multigraph λKn. Theorem 3 ([4]). Let C denote a collection of r directed cycles of lengths m1, m2, . We will let ν = ν2(M ) denote the number of cycles of length 2 in the list M. In Lemma 23 we exhibit a family of n-admissible lists for which Kn∗ admits no M -decomposition
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