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. dc(v) denotes the color degree of a vertex v of G. In a previous paper, we showed that if dc(v) ≥ k for every vertex v of G, then G has a heterochromatic path of length at least dk+1 2 e. 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 d2k+1 3 e. 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 d3k 5 e + 1. Actually, we can show that for 1 ≤ k ≤ 5 any graph G under the color degree condition has a heterochromatic path of length at least k, with only one exceptional graph K4 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.