Color degree condition for long rainbow paths in edge - colored graphs ∗

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 by dc(v). In a previous paper, we showed that if dc(v) ≥ k (color degree condition) for every vertex v of G, then G has a rainbow path of length at least d(k + 1)/2e. Later, in another paper we first showed that if k ≤ 7, G has a rainbow path of length at least k − 1, and then, based on this we used induction on k and showed that if k ≥ 8, then G has a rainbow path of length at least d(3k)/5e+1. In 2010, Gyárfás and Mhalla showed that in any proper edge-colored complete graph Kn, there is a rainbow path with no less than (2n + 1)/3 vertices. In the present paper, by using a simpler approach we further improve the result by showing that if k ≥ 8, G has a rainbow path of length at least d(2k)/3e+ 1.