Decomposing edge-colored graphs under color degree constraints

For an edge-colored graph G, the minimum color degree of G means the minimum number of colors on edges which are incident to each vertex of G. We prove that if G is an edge-colored graph with minimum color degree at least 5 then V (G) can be partitioned into two parts such that each part induces a subgraph with minimum color degree at least 2. We show this theorem by proving a much stronger form. Moreover, we point out an important relationship between our theorem and Bermond-Thomassen's conjecture in digraphs.