A Note on Heterochromatic C 4 in Edge-Colored Triangle-Free Graphs

Abstract

Let G be an edge-colored graph. A heterochromatic cycle of G is a cycle in which any pair of edges have distinct colors. Let dc(v), named the color degree of a vertex v, be defined as the maximum number of edges incident with v, that have distinct colors. In this paper, we prove that if G is an edge-colored triangle-free graph of order n ≥ 9 and \({d^c(v) \geq \frac{(3-\sqrt{5})n}{2}+1}\) for each vertex v of G, G has a heterochromatic C4.