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 CN(v) denote the color neighborhood of a vertex v of G. In a previous paper, we showed that 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 b 2s+4 5 c. In the present paper, we prove that G has a heterochromatic path of length at least d s+1 2 e, and give examples to show that the lower bound is best possible in some sense.