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تسجيل مستخدم جديدOn Even Rainbow or Nontriangular Directed Cycles
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Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We establish a corresponding result for fixed even rainbow $ell$-cycles $C_{ell}$: if every vertex $v in V$ is incident to at least $(n+5)/3$ distinctly colored edges, where $n geq n_0(ell)$ is sufficiently large, then $G$ admits an even rainbow $ell$-cycle $C_{ell}$. This result is best possible whenever $ell otequiv 0$ (mod 3). Correspondingly, we also show that for a fixed (even or odd) integer $ell geq 4$, every large $n$-vertex oriented graph $vec{G} = (V, vec{E})$ with minimum outdegree at least $(n+1)/3$ admits a (consistently) directed $ell$-cycle $vec{C}_{ell}$. Our latter result relates to one of Kelly, Kuhn, and Osthus, who proved a similar statement for oriented graphs with large semi-degree. Our proofs are based on the stability method.
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