Abstract
Let Gϕ be an edge-colored graph with order n and δGϕ be the smallest number of distinctly colored edges incident to a vertex of Gϕ. To determine the smallest function h(n) such that any edge-colored graph Gϕ of order n contains a rainbow cycle if δGϕ≥h(n)? It is known that log2n<h(n)≤n+12. For a positive integer r, an edge-coloring ϕ of Gϕ is called r-good if each color appears at most r times at each vertex to generalize the proper edge-coloring of Gϕ. In this paper, for any given ϵ ∈ (0, 1), we show that there exists a constant C(ϵ,r) such that h(n)<C(ϵ,r)nϵ for any r-good edge-colored graph Gϕ.
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