In a simple, undirected graph G, an edge 2-coloring is a coloring of the edges such that no vertex is incident to edges with more than 2 distinct colors. The problem maximum edge 2-coloring (ME2C) is to find an edge 2-coloring in a graph G with the goal to maximize the number of colors. For a relevant graph class, ME2C models anti-Ramsey numbers and it was considered in network applications. For the problem a 2-approximation algorithm is known, and if the input graph has a perfect matching, the same algorithm has been shown to have a performance guarantee of 5/3. It is known that ME2C is APX-hard and that it is UG-hard to obtain an approximation ratio better than 1.5. We show that if the input graph has a perfect matching, there is a polynomial time 1.625-approximation and if the graph is claw-free or if the maximum degree of the input graph is at most three (i.e., the graph is subcubic), there is a polynomial time 1.5-approximation algorithm for ME2C

翻译：在简单无向图G中，边2-着色是指对边进行着色，使得每个顶点关联的边至多包含2种不同颜色。最大边2-着色问题（ME2C）的目标是寻找图G的一种边2-着色，使得使用的颜色数量最大化。针对相关图类，ME2C可建模反拉姆齐数，并已在网络应用中得到研究。该问题已知存在2-近似算法，且当输入图包含完美匹配时，该算法的性能保证可改进为5/3。已知ME2C是APX-hard问题，且若近似比优于1.5则属于UG-hard问题。我们证明：当输入图包含完美匹配时，存在多项式时间1.625-近似算法；若图是无爪图或输入图的最大度不超过3（即次立方图），则存在多项式时间1.5-近似算法。