Long-heterochromatic paths in edge-colored graphs

Let G be an edge-colored graph. A heterochromatic path of G is such a path in which no two edges have the same color. d(c)(v) denotes the color degree of a vertex v of G. In a previous paper, we showed that if d(c)(v) >= k for every vertex v of G, then G has a heterochromatic path of length at least [k+1/2]. It is easy to see that if k = 1, 2, G has a heterochromatic path of length at least k. Saito conjectured that under the color degree condition G has a heterochromatic path of length at least [2k+1/3]. Even if this is true, no one knows if it is a best possible lower bound. Although we cannot prove Saito's conjecture, we can show in this paper that if 3 <= k <= 7, G has a heterochromatic path of length at least k - 1, and if k >= 8, G has a heterochromatic path of length at least [3k/5] + 1. Actually, we can show that for t <= k <= 5 any graph G under the color degree condition has a heterochromatic path of length at least k, with only one exceptional graph K-4 for k = 3, one exceptional graph for k = 4 and three exceptional graphs for k = 5, for which G has a heterochromatic path of length at least k - 1. Our experience suggests us to conjecture that under the color degree condition G has a heterochromatic path of length at least k - 1.