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{\bf Maria Axenovich and JiHyeok Choi}
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{\bf On Colorings Avoiding a Rainbow Cycle and a Fixed Mono\-chromatic Subgraph}
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Let $H$ and $G$ be two graphs on fixed number of
vertices. An edge coloring of a complete graph is called
$(H,G)$-good if there is no monochromatic copy of $G$ and no
rainbow (totally multicolored) copy of $H$ in this coloring. As
shown by Jamison and West, an $(H,G)$-good coloring
of an arbitrarily large complete graph exists unless either $G$
is a star or $H$ is a forest. The largest number of colors in
an $(H,G)$-good coloring of $K_n$ is denoted $maxR(n, G,H)$.
For graphs $H$ which can not be vertex-partitioned into at most
two induced forests, $maxR(n, G,H)$ has been determined
asymptotically. Determining $maxR(n; G, H)$ is
challenging for other graphs $H$, in particular for bipartite
graphs or even for cycles. This manuscript treats the case when
$H$ is a cycle. The value of $maxR(n, G, C_k)$ is determined
for all graphs $G$ whose edges do not induce a star.

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