Let \(G=(V,E,c)\) be an edge-colored graph, where \(c:E\rightarrow \mathbb {N}\) is an edge coloring of G. Let \(d_G^{c}(v)\) be the color degree of v, which is the number of distinct colors on incident edges of v. Let \(\delta ^{c}(G)=\min \{d_G^{c}(v)~|~v\in V(G)\}\) be the minimum color degree of G (with respect to c). Let s, t be two integers with \(s\ge t\ge 2\) and G be an edge-colored graph with \(\delta ^{c}(G)\ge s+t+1\) . Fujita, Li and Wang in 2019 conjectured that G admits a partition (S, T) such that \(\delta ^{c}(G[S])\ge s\) and \(\delta ^{c}(G[T])\ge t\) . Here G[U] denotes the subgraph of G induced by the vertex set U. Let \(E^{i}=\{e\in E(G)~|~c(e)=i\}\) be the color class for color i, let \(G^{i}=(V(G),E^{i})\) be the corresponding subgraph. A bipartite graph \(K_{1,3}\) is called a claw, and a graph is claw-free if it does not contain a claw as an induced subgraph. We say an edge-colored graph G is a (monochromatic claw)-free graph if for each \(i\in \mathbb {N}\) , \(G^{i}\) is claw-free. In this paper, we first show that a (monochromatic claw)-free graph G admits a partition (S, T) such that \(\delta ^{c}(G[S])\ge s\) and \(\delta ^{c}(G[T])\ge t\) if \(\delta ^{c}(G)\ge 2s+t+1\) . A monochromatic path of length 2 is referred to as a monochromatic 2-path. Let G be an edge-colored graph in which no end-vertex of a monochromatic 2-path lies on another monochromatic 2-path of a different color. Then, we show that such an edge-colored graph G admits a partition (S, T) such that \(\delta ^{c}(G[S])\ge s\) and \(\delta ^{c}(G[T])\ge t\) if \(\delta ^{c}(G)\ge s+t+1\) .
