Rainbow path and color degree in edge colored graphs

Let G be an edge colored graph. A 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 (2005) showed that if d(c) >= k (k >= 8), for every vertex v of C, then t >= inverted left perpendicular 3k/5inverted right perpendicular + 1. Unfortunately, the 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 >= inverted left perpendicular 2k/3inverted right perpendicular. They also state in their paper that they believe t >= k - c for some constant c. In this note, we give a short proof to show that t >= inverted left perpendicular 3k/5inverted right perpendicular, 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.