# Long rainbow path in proper edge colored complete graphs

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He Chen

Department of Mathematics, Southeast University, China

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Xueliang Li

Center for Combinatorics, Nankai University, China

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**Minisymposium:**
VERTEX COLOURINGS AND FORBIDDEN SUBGRAPHS

**Content:**
Let $G$ be an edge-colored graph. A rainbow (heterochromatic, or
multicolored) path of $G$ is such a path in which no two edges have
the same color. Let the color degree of a vertex $v$ to be the
number of different colors that are used on edges incident to $v$,
and denote it to be $d^c(v)$. It was showed that if $d^c(v)\geq k$
$v$ of $G$, then $G$ has a rainbow path of length at least
$\min\{\lceil\frac{2k+1}{3}\rceil,k-1\}$. In the present paper, we
consider proper edge-colored complete graph $K_n$ only and improve
the lower bound of the length of the longest rainbow path by showing
that if $n\geq 20$, there must have a rainbow path of length no less
than $\displaystyle
\frac{3}{4}n-\frac{1}{4}\sqrt{\frac{n}{2}-\frac{39}{11}}-\frac{11}{16}$.