Color degree and color neighborhood union conditions for long heterochromatic paths in edge - colored graphs ∗

Let G be an edge-colored graph. A heterochromatic (rainbow, or multicolored) path of G is such a path in which no two edges have the same color. Let d(v) denote the color degree and CN(v) denote the color neighborhood of a vertex v of G. In a previous paper, we showed that if d(v) ≥ k (color degree condition) for every vertex v of G, then G has a heterochromatic path of length at least ⌈ 2 ⌉, and if |CN(u) ∪ CN(v)| ≥ s (color neighborhood union condition) for every pair of vertices u and v of G, then G has a heterochromatic path of length at least ⌈ s 3⌉ + 1. Later, in another paper we first showed that if k ≤ 7, G has a heterochromatic path of length at least k− 1, and then, based on this we use induction on k and showed that if k ≥ 8, then G has a heterochromatic path of length at least ⌈ 5 ⌉ + 1. In the present paper, by using a simpler approach we further improve the result by showing that if k ≥ 8, G has a heterochromatic path of length at least ⌈ 3 ⌉+ 1, which confirms a conjecture by Saito. We also improve a previous result by showing that under the color neighborhood union condition, G has a heterochromatic path of length at least ⌊ 5 ⌋.