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 C N (v) denote the color neighborhood of a vertex v of G. In a previous paper, we showed that if vertical bar C N (u) boolean OR C N (v)vertical bar >= 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 [2s+4/5]. In the present paper, we prove that G has a heterochromatic path of length at least [s+1/2], and give examples to show that the lower bound is best possible in some sense.