Rainbow path and color degree in edge colored graphs

Let $G$ be an edge colored graph. A {\it}{rainbow path} in $G$ is a path in which all the edges are colored with distinct colors. Let $d^c(v)$ be the color degree of a vertex $v$ in $G$, i.e. the number of distinct colors present on the edges incident on the vertex $v$. Let $t$ be the maximum length of a rainbow path in $G$. Chen and Li showed that if $d^c \geq k$, for every vertex $v$ of $G$, then $t \geq \left \lceil \frac{3 k}{5}\right \rceil + 1$ (Long heterochromatic paths in edge-colored graphs, The Electronic Journal of Combinatorics 12 (2005), # R33, Pages:1-33.) Unfortunately, proof by Chen and Li is very long and comes to about 23 pages in the journal version. Chen and Li states in their paper that it was conjectured by Akira Saito, that $t \ge \left \lceil \frac {2k} {3} \right \rceil$. They also states in their paper that they believe $t \ge k - c$ for some constant $c$. In this note, we give a short proof to show that $t \ge \left \lceil \frac{3 k}{5}\right \rceil$, using an entirely different method. Our proof is only about 2 pages long. The draw-back is that our bound is less by 1, than the bound given by Chen and Li. We hope that the new approach adopted in this paper would eventually lead to the settlement of the conjectures by Saito and/or Chen and Li.Comment: 4 page

The minimum color degree and a large rainbow cycle in an edge-colored graph

Let $G$ be an edge-colored graph with $n$ vertices. A subgraph $H$ of $G$ is called a rainbow subgraph of $G$ if the colors of each pair of the edges in $E(H)$ are distinct. We define the minimum color degree of $G$ to be the smallest number of the colors of the edges that are incident to a vertex $v$, for all $v\in V(G)$. Suppose that $G$ contains no rainbow-cycle subgraph of length four. We show that if the minimum color degree of $G$ is at least $\frac{n+3k-2}{2}$, then $G$ contains a rainbow-cycle subgraph of length at least $k$, where $k\geq 5$. Moreover, if the condition of $G$ is restricted to a triangle-free graph that contains a rainbow path of length at least $\frac{3k}{2}$, then the lower bound of the minimum color degree of $G$ that guarantees an existence of a rainbow-cycle subgraph of length to at least $k$ can be reduced to $\frac{2n+3k-1}{4}$.Comment: 8 page