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- W2949251486 abstract "For a given graph $H$ and $ngeq 1$, let $f(n,H)$ denote the maximum number $c$ for which there is a way to color the edges of the complete graph $K_n$ with $c$ colors such that every subgraph $H$ of $K_n$ has at least two edges of the same color. Equivalently, any edge-coloring of $K_n$ with at least $rb(n,H)=f(n,H)+1$ colors contains a rainbow copy of $H$, where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number $rb(n,H)$ is called the {it rainbow number of $H$}. ErdH{o}s, Simonovits and S'{o}s showed that $rb(n,K_3)=n$. In 2004, Schiermeyer used some counting technique and determined the rainbow numbers $rb(n,kK_2)$ for $kgeq 2$ and $ngeq 3k+3$. It is easy to see that $n$ must be at least $2k$. So, for $2k leq n<3k+3$, the rainbow numbers remain not determined. In this paper we will use the Gallai-Edmonds structure theorem for matchings to determine the exact values for rainbow numbers $rb(n,kK_2)$ for all $kgeq 2$ and $ngeq 2k$, giving a complete solution for the rainbow numbers of matchings." @default.
- W2949251486 created "2019-06-27" @default.
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- W2949251486 date "2006-11-16" @default.
- W2949251486 modified "2023-09-27" @default.
- W2949251486 title "Complete Solution for the Rainbow Numbers of Matchings" @default.
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