Long rainbow path in proper edge colored complete graphs

He Chen
Department of Mathematics, Southeast University, China

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 $\displaystyle \frac{3}{4}n-\frac{1}{4}\sqrt{\frac{n}{2}-\frac{39}{11}}-\frac{11}{16}$.

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